Appendix to the article “On the number of Descendants and Ascendants in Random Search Trees” by C. Martínez, A. Panholzer, and H. Prodinger

Appendix A. Polynomials

We collect here various polynomials in two and three variables that appear as factors or terms in the explicit form of some the generating functions studied in this paper.

A.1. The polynomials A_{i}(z, u, v).

▷ A_{0}(z, u, v) = \frac{1}{70}(3410v^{7}u^{6}z^{6}+7356u^{4}v^{4}z+900v^{8}u^{8}z^{7}−1316u^{9}v^{7}z^{5}−1148u^{9}v^{8}z^{6}−880v^{7}u^{8}z^{7}−

210u^{9}v^{8}z^{4}−5874v^{7}u^{5}z^{5}+4886v^{7}u^{8}z^{4}+4080v^{4}u^{6}z−2564v^{8}u^{8}z^{6}−4648v^{5}u^{3}z+140u^{10}v^{8}z^{7}+3822u^{5}v^{8}z^{5}− 2378u^{3}v^{3}z − 680u^{9}v^{7}z^{7} + 7356v^{4}u^{5}z − 120vu^{2}z − 880v^{7}u^{7}z^{7} + 1388v^{7}u^{9}z^{6} − 490v^{9}u^{5}z + 1042v^{4}u^{2}z + 3484u^{7}v^{7}z^{6}+120u^{5}z^{2}+60zu^{3}−120z^{2}u^{4}−140u^{6}v^{7}−140u^{4}v^{8}+70u^{5}v^{8}+140u^{6}v^{6}+70v^{2}u^{3}−140u^{5}v^{4}− 140v^{6}u^{4}+140u^{5}v^{3}−140u^{4}v^{4}−140v^{4}u^{3}−140v^{2}u^{4}+60uv^{2}z+10780v^{7}u^{6}z^{2}−130v^{8}u^{8}z^{8}−3920u^{6}v^{8}z^{2}+ 4080u^{3}v^{4}z−5874u^{8}v^{7}z^{5}−1378v^{5}u^{2}z−4648v^{5}u^{6}z−9220v^{5}u^{5}z−9220v^{5}u^{4}z+12362v^{7}u^{5}z^{4}+60v^{7}u^{8}z^{8}− 80v^{8}u^{9}z^{8} +130u^{9}v^{7}z^{8} −45u^{10}v^{8}z^{8} +3410u^{8}v^{7}z^{6} +3822u^{8}v^{8}z^{5} −8606v^{7}u^{7}z^{5} −70v^{9}u^{9}z^{4} −84v^{8}u^{10}z^{6} + 1512v^{9}u^{7}z^{6}+868v^{9}u^{8}z^{6}+6986v^{8}z^{5}u^{6}+1106u^{9}v^{8}z^{5}+868v^{9}u^{6}z^{6}−400v^{9}u^{7}z^{7}+900v^{8}u^{7}z^{7}−2564v^{8}u^{6}z^{6}− 2590v^{9}u^{7}z^{5}+70v^{9}u^{8}z^{8}−3560v^{8}u^{7}z^{6}−826v^{9}u^{8}z^{5}−2590v^{9}u^{6}z^{5}−826v^{9}u^{5}z^{5}−2800v^{9}u^{5}z^{3}−400v^{9}u^{8}z^{7}− 5v^{9}u^{10}z^{8}−100v^{9}u^{9}z^{7}+1806v^{5}u^{6}z^{6}−2370v^{5}u^{5}z^{5}−520u^{3}v^{2}z^{2}+210uv^{6}z^{4}−14104v^{4}u^{5}z^{2}+56u^{10}v^{7}z^{5}− 220v^{5}z^{7}u^{6} −1612u^{4}v^{2}z^{2} +110v^{3}u^{2}z^{2} −1612u^{6}v^{2}z^{2} −1246v^{6}u^{9}z^{6} +8400u^{6}v^{5}z^{4} +140u^{5}v^{7} +140v^{3}u^{3} − 140v^{6}u^{3} +140v^{4}u^{2} +70v^{4}u−140v^{3}u^{2} −140u^{5}v^{5} −140v^{5}u−140v^{5}u^{3} −140u^{5}v^{6} +140u^{6}v^{4} +140v^{7}u^{3} + 560v^{5}u^{4} − 140v^{3}u^{6} + 70v^{2}u^{5} + 60u^{6}z + 140v^{6}u^{2} − 140v^{7}u^{2} + 70uv^{6} + 70v^{8}u^{3} − 140v^{5}u^{7} + 70u^{7}v^{4} + 42v^{9}u^{9}z^{5} − 126v^{8}u^{10}z^{5} + 14v^{9}u^{10}z^{5} + 84v^{9}u^{9}z^{6} + 30v^{9}u^{9}z^{8} − 28v^{9}u^{10}z^{6} − 4062v^{3}u^{4}z − 4062v^{3}u^{5}z + 7732u^{5}v^{6}z+770u^{6}v^{8}z+7732v^{6}u^{4}z+11760v^{7}u^{5}z^{2}−4620v^{7}u^{5}z−140v^{9}u^{7}z^{2}−4620v^{7}u^{4}z+6986v^{8}u^{7}z^{5}+ 1890v^{8}u^{5}z + 1534v^{2}u^{4}z − 8606v^{7}u^{6}z^{5} + 1960v^{9}u^{5}z^{2} − 2800v^{9}u^{6}z^{3} + 9240v^{8}u^{6}z^{3} − 1260v^{8}u^{7}z^{2} − 15232u^{6}v^{7}z^{3} + 12362v^{7}u^{7}z^{4} − 3038v^{8}u^{8}z^{4} − 7000v^{8}u^{5}z^{2} + 2170v^{9}u^{5}z^{4} + 3780v^{9}u^{6}z^{4} − 7490v^{8}u^{5}z^{4} + 70v^{9}u^{6}z−580u^{2}v^{3}z+360u^{2}v^{2}z−15232v^{7}u^{5}z^{3}−700v^{9}u^{7}z^{3}−180vzu^{3}+746v^{2}zu^{3}+122v^{4}zu−180v^{3}uz+ 9240v^{8}u^{5}z^{3} + 980v^{8}u^{8}z^{3} + 210v^{9}u^{8}z^{4} + 2170v^{9}u^{7}z^{4} + 13888v^{7}u^{6}z^{4} − 7490v^{8}u^{7}z^{4} − 10724v^{8}u^{6}z^{4} + 840v^{9}u^{6}z^{2}−9464v^{7}u^{7}z^{3}−2520v^{7}u^{6}z+4252v^{6}u^{6}z−1148u^{5}v^{8}z^{6}+2800u^{3}v^{7}z^{2}+126v^{7}u^{2}z^{5}−3038u^{4}v^{8}z^{4}+ 1106u^{4}v^{8}z^{5} + 826u^{2}v^{6}z − 2520v^{7}u^{3}z + 10v^{8}u^{5}z^{8} + 500u^{6}v^{8}z^{7} − 680v^{7}u^{6}z^{7} + 140v^{8}u^{5}z^{7} + 20v^{9}u^{5}z^{7} − 28v^{9}u^{4}z^{6} − 210uv^{6}z − 1316v^{7}u^{4}z^{5} − 100v^{7}u^{5}z^{7} + 25v^{7}u^{6}z^{8} + 112v^{7}u^{4}z^{6} − 126v^{8}u^{3}z^{5} + 56v^{7}u^{3}z^{5} + 42v^{9}u^{4}z^{5}+14v^{9}u^{3}z^{5}−5v^{9}u^{6}z^{8}−84v^{8}u^{4}z^{6}−84v^{7}u^{3}z^{6}−210v^{7}u^{2}z^{4}−210v^{8}u^{3}z^{4}+56v^{8}u^{3}z^{6}−28v^{8}u^{2}z^{5}− 45v^{8}u^{6}z^{8}+1388v^{7}u^{5}z^{6}−70v^{9}u^{3}z^{4}−180u^{6}vz−140u^{2}v^{8}z+70uv^{7}z+280u^{2}v^{8}z^{2}−70u^{2}v^{7}z+140z^{4}u^{2}v^{8}+ 126v^{5}uz + 140v^{7}u^{2}z^{3} − 2378u^{6}v^{3}z + 448v^{7}u^{3}z^{4} + 746u^{6}v^{2}z − 300u^{5}vz + 4886v^{7}u^{4}z^{4} − 100v^{9}u^{6}z^{7} + 84v^{9}u^{5}z^{6} + 140v^{9}u^{3}z^{3} + 210v^{9}u^{4}z^{4} + 980v^{8}u^{3}z^{3} − 1624v^{7}u^{3}z^{3} + 1534u^{5}v^{2}z − 300u^{4}vz − 700v^{9}u^{4}z^{3} − 1260u^{3}v^{8}z^{2}−140u^{3}v^{9}z^{2}+4252v^{6}u^{3}z+20v^{7}u^{4}z^{7}+4340v^{8}u^{4}z^{3}−9464v^{7}u^{4}z^{3}+872v^{5}u^{8}z^{4}+176v^{4}u^{8}z^{4}+ 7824v^{5}u^{7}z^{4} − 100v^{7}u^{10}z^{7} + 6554u^{7}v^{6}z^{5} + 3998u^{8}v^{6}z^{5} + 14940u^{4}v^{5}z^{2} + 1520v^{3}u^{6}z^{4} + 2246v^{4}u^{7}z^{5} − 9220u^{4}v^{4}z^{2}−46u^{8}v^{4}z^{5}+320v^{4}z^{7}u^{7}+1280v^{6}u^{9}z^{5}+380v^{6}u^{9}z^{7}−2804u^{5}v^{3}z^{3}+1806v^{5}u^{8}z^{6}+540u^{5}vz^{2}+ 120v^{6}u^{10}z^{7}−526v^{5}u^{9}z^{6}−286v^{4}u^{8}z^{6}−140v^{7}uz^{2}+840u^{4}v^{9}z^{2}+770u^{3}v^{8}z−13520u^{6}v^{5}z^{3}+14940v^{5}u^{6}z^{2}+ 252u^{2}v^{6}z^{2}+420v^{6}uz^{2}+7880v^{4}u^{5}z^{3}+9428v^{6}u^{7}z^{3}−1344u^{9}v^{6}z^{3}+412u^{8}v^{5}z^{3}+2940v^{6}u^{8}z^{3}−5014v^{6}u^{8}z^{4}− 2416v^{4}u^{7}z^{4}+1302v^{6}u^{10}z^{4}+2246v^{4}z^{5}u^{6}−1834v^{6}z^{6}u^{6}+460v^{3}u^{5}z^{4}−5200v^{4}u^{6}z^{4}−70v^{7}u^{7}z+2800v^{7}u^{7}z^{2}+ 1890v^{8}u^{4}z+448v^{7}u^{9}z^{4}−210v^{7}u^{10}z^{4}−2416v^{4}u^{5}z^{4}−3920v^{8}u^{4}z^{2}+336v^{6}u^{9}z^{4}−1876v^{5}u^{9}z^{4}+1560v^{5}u^{7}z^{6}− 490v^{9}u^{4}z −8008v^{5}u^{7}z^{3} +2948u^{7}v^{4}z^{3} +540u^{5}z^{3}v^{2} +150u^{4}z^{2}v+70v^{9}u^{3}z −5964v^{6}u^{3}z^{2} −2804u^{6}v^{3}z^{3} + 7880u^{6}v^{4}z^{3} + 4856v^{3}u^{4}z^{2} + 4336v^{5}u^{3}z^{2} + 6552u^{5}v^{3}z^{2} + 252v^{6}u^{8}z^{2} + 4336v^{5}u^{7}z^{2} − 1378v^{5}u^{7}z − 12914v^{6}u^{4}z^{2} − 12914v^{6}u^{6}z^{2} + 17336v^{5}u^{5}z^{2} + 1042u^{7}v^{4}z − 3634u^{7}v^{4}z^{2} − 580u^{7}v^{3}z + 1658u^{7}v^{3}z^{2} + 360u^{7}v^{2}z + 196u^{8}v^{5}z^{2} − 19028u^{5}v^{6}z^{2} − 9220u^{6}v^{4}z^{2} + 4856u^{6}v^{3}z^{2} − 2096u^{5}v^{2}z^{2} − 4366v^{5}z^{5}u^{6} + 432v^{6}u^{10}z^{6}+3998v^{6}z^{5}u^{5}+1812v^{5}z^{5}u^{4}+500v^{8}u^{9}z^{7}+112u^{10}v^{7}z^{6}−220u^{9}v^{5}z^{7}+308v^{4}z^{6}u^{5}−100v^{4}z^{7}u^{6}+ 872v^{5}u^{4}z^{4} − 1246v^{6}z^{6}u^{5} + 6554v^{6}z^{5}u^{6} + 17116v^{6}u^{6}z^{3} + 1280v^{6}z^{5}u^{4} + 7824v^{5}u^{5}z^{4} − 700v^{4}z^{5}u^{4} − 46v^{4}z^{5}u^{5}−286v^{4}z^{6}u^{6}−1344v^{6}u^{10}z^{5}−9772u^{7}v^{6}z^{4}−1834u^{8}v^{6}z^{6}−546v^{6}z^{5}u^{2}+460u^{5}v^{5}z^{7}−70v^{7}uz^{4}− 952v^{5}u^{4}z^{6}−4366u^{7}v^{5}z^{5}+120v^{6}u^{5}z^{7}−2370u^{8}v^{5}z^{5}−1344u^{3}v^{6}z^{5}−300u^{4}v^{6}z^{7}+1302u^{2}v^{6}z^{4}+588u^{3}v^{6}z^{6}+ 1148u^{3}v^{5}z^{5} + 432u^{4}v^{6}z^{6} + 10780v^{7}u^{4}z^{2} − 13724u^{6}v^{6}z^{4} − 2648u^{7}v^{6}z^{6} − 526v^{5}z^{6}u^{5} − 5964v^{6}u^{7}z^{2} + 1812u^{9}v^{5}z^{5}−34v^{4}z^{2}u^{2}−1396v^{4}u^{7}z^{6}+320v^{4}u^{8}z^{7}+4340v^{8}u^{7}z^{3}−3634v^{4}u^{3}z^{2}+196v^{5}u^{2}z^{2}+2940u^{3}v^{6}z^{3}+ 412v^{5}u^{3}z^{3}−1344v^{6}u^{2}z^{3}−8008u^{4}v^{5}z^{3}−13520u^{5}v^{5}z^{3}−88v^{4}u^{3}z^{3}−856v^{3}u^{4}z^{3}+9428v^{6}u^{4}z^{3}−1624v^{7}u^{8}z^{3}+ 17116v^{6}u^{5}z^{3} +1658u^{3}v^{3}z^{2} −500v^{5}u^{8}z^{7} +140v^{7}u^{9}z^{3} −420v^{5}z^{2}u+140v^{7}uz^{3} −280u^{2}v^{8}z^{3} −420v^{6}uz^{3} + 336u^{3}v^{6}z^{4}−1876v^{5}u^{3}z^{4}−5014v^{6}u^{4}z^{4}−910v^{5}z^{4}u^{2}+896v^{5}z^{3}u^{2}+2948v^{4}u^{4}z^{3}−9772v^{6}u^{5}z^{4}+826v^{6}u^{7}z+ 980v^{4}u^{3}z^{4} + 176v^{4}u^{4}z^{4} + 126v^{5}u^{8}z + 122u^{8}v^{4}z − 180u^{8}v^{3}z + 60u^{8}v^{2}z − 420u^{9}v^{5}z^{2} + 420v^{6}u^{9}z^{2} + 210v^{6}u^{11}z^{4} + 70v^{7}u^{8}z − 70v^{7}u^{11}z^{4} − 420u^{10}v^{6}z^{3} − 120u^{7}vz − 280v^{8}u^{9}z^{3} + 140v^{9}u^{8}z^{3} − 28v^{8}u^{11}z^{5} + 20v^{9}u^{10}z^{7} + 56v^{8}u^{11}z^{6} + 25u^{10}v^{7}z^{8} + 10u^{11}v^{8}z^{8} − 40u^{11}v^{8}z^{7} + 140u^{10}v^{8}z^{4} + 70u^{7}v^{6} − 5v^{7}u^{4}z^{8} + 20u^{6}z^{7}v^{2} + 14v^{7}u^{12}z^{5} + 20v^{7}u^{12}z^{7} + 15v^{6}u^{12}z^{8} − 5v^{7}u^{12}z^{8} − 20v^{4}u^{12}z^{7} + 28v^{4}u^{12}z^{6} + 60v^{5}u^{12}z^{7} + 5v^{4}u^{12}z^{8} − 15v^{5}u^{12}z^{8} − 42v^{6}u^{12}z^{5} + 84v^{6}u^{12}z^{6} − 28v^{7}u^{12}z^{6} − 60v^{6}u^{12}z^{7} + 42v^{5}u^{12}z^{5} + 140v^{4}u^{9}z^{2} + 420v^{5}u^{10}z^{3}−210v^{5}u^{11}z^{4}−14v^{4}u^{12}z^{5}−84v^{5}u^{12}z^{6}−140v^{4}u^{10}z^{3}+70v^{4}u^{11}z^{4}+40u^{11}v^{3}z^{7}−10v^{3}z^{8}u^{11}+ 5v^{2}z^{8}u^{10} − 56u^{11}v^{3}z^{6} + 280u^{9}v^{3}z^{3} − 140u^{10}v^{3}z^{4} + 70z^{4}u^{9}v^{2} + 28u^{10}z^{6}v^{2} − 20u^{10}z^{7}v^{2} − 14u^{10}z^{5}v^{2} + 28u^{11}v^{3}z^{5} − 210v^{6}u^{8}z + 240u^{7}vz^{2} − 120u^{8}z^{2}v^{2} − 140z^{3}u^{8}v^{2} + 140v^{7}u^{10}z^{3} − 140v^{7}u^{9}z^{2} − 60u^{5}vz^{3} + 150u^{6}vz^{2} + 240u^{3}vz^{2} − 520u^{7}z^{2}v^{2} + 98z^{5}u^{8}v^{2} + 260z^{3}u^{7}v^{2} + 70z^{4}u^{8}v^{2} + 14z^{5}u^{7}v^{2} − 84z^{6}u^{8}v^{2} − 210z^{4}u^{7}v^{2} + 28u^{9}z^{6}v^{2} + 20u^{9}z^{7}v^{2} − 98u^{9}z^{5}v^{2} + 238u^{10}v^{3}z^{5} − 70u^{8}v^{3}z^{4} + 154u^{9}v^{3}z^{5} − 140u^{10}v^{3}z^{6} + 80u^{8}v^{3}z^{3} − 350u^{9}v^{3}z^{4} − 856u^{7}v^{3}z^{3} + 540u^{6}v^{2}z^{3} + 460u^{7}v^{3}z^{4} + 20u^{6}v^{2}z^{4} + 70u^{8}v^{3}z^{5} − 28u^{9}v^{3}z^{6} − 550v^{3}z^{5}u^{7} + 14v^{2}z^{5}u^{6} + 28v^{3}z^{6}u^{8} + 56v^{2}z^{6}u^{7} + 20u^{10}v^{3}z^{7} + 20v^{3}z^{7}u^{9} + 5v^{3}z^{8}u^{10} − 10v^{2}z^{8}u^{9} + 98v^{2}z^{5}u^{5} − 84v^{2}z^{6}u^{6} − 210v^{2}z^{4}u^{5} + 10v^{2}z^{8}u^{8} − 550v^{3}u^{6}z^{5} + 392v^{3}u^{7}z^{6} + 70v^{3}z^{5}u^{5} + 28v^{3}z^{6}u^{6} − 80v^{3}u^{8}z^{7} − 10v^{3}u^{9}z^{8} − 80v^{3}z^{7}u^{7} + 30v^{3}z^{8}u^{8} − 88v^{4}u^{8}z^{3} + 308v^{4}u^{9}z^{6} + 392v^{4}u^{10}z^{6} − 100v^{4}u^{9}z^{7} − 700v^{4}u^{9}z^{5} −644v^{4}u^{10}z^{5} −100v^{4}u^{10}z^{7} −700v^{4}u^{9}z^{3} +490v^{4}u^{10}z^{4} +980v^{4}u^{9}z^{4} +5v^{4}z^{8}u^{10} −34v^{4}u^{8}z^{2} + 1148v^{5}u^{10}z^{5} + 896v^{5}u^{9}z^{3} − 910v^{5}u^{10}z^{4} − 952v^{5}u^{10}z^{6} − 182v^{4}u^{11}z^{5} − 364v^{5}u^{11}z^{6} − 100v^{4}u^{11}z^{7} + 434v^{5}u^{11}z^{5}+196v^{4}u^{11}z^{6}+140v^{5}u^{11}z^{7}+20v^{4}u^{11}z^{8}+460v^{5}u^{10}z^{7}−20v^{5}u^{11}z^{8}−95v^{5}z^{8}u^{10}+20v^{4}z^{8}u^{9}+ 520v^{6}u^{8}z^{7} −100v^{4}z^{8}u^{8} +120v^{5}u^{9}z^{8} −500v^{5}z^{7}u^{7} +20v^{5}z^{8}u^{8} −75v^{6}z^{8}u^{10} −546v^{6}u^{11}z^{5} +588v^{6}u^{11}z^{6} − 84v^{7}u^{11}z^{6} − 300v^{6}u^{11}z^{7} + 126v^{7}u^{11}z^{5} + 20v^{7}u^{11}z^{7} + 60v^{6}u^{11}z^{8} − 20v^{6}z^{8}u^{9} + 520v^{6}z^{7}u^{7} − 80v^{6}z^{8}u^{8} + 14v^{7}uz^{5} − 182v^{4}u^{2}z^{5} − 644u^{3}v^{4}z^{5} − 364v^{5}u^{3}z^{6} + 196v^{4}u^{3}z^{6} + 420uv^{5}z^{3} + 490v^{4}u^{2}z^{4} − 14v^{4}z^{5}u + 70v^{4}uz^{4}−84v^{5}u^{2}z^{6}−700u^{2}v^{4}z^{3}−140uv^{4}z^{3}+140uv^{4}z^{2}+42v^{5}z^{5}u−40v^{8}u^{4}z^{7}+140u^{4}v^{5}z^{7}−42v^{6}z^{5}u+ 434u^{2}v^{5}z^{5} + 84u^{2}v^{6}z^{6} + 28v^{4}u^{2}z^{6} − 20v^{4}u^{3}z^{7} + 60v^{5}u^{3}z^{7} − 120u^{2}z^{2}v^{2} − 140u^{3}z^{3}v^{2} − 14z^{5}u^{3}v^{2} + 70z^{4}u^{3}v^{2}+28u^{4}z^{6}v^{2}+28u^{5}z^{6}v^{2}−98u^{4}z^{5}v^{2}−20u^{5}z^{7}v^{2}+260u^{4}z^{3}v^{2}+70u^{4}z^{4}v^{2}+20v^{7}u^{3}z^{7}−28u^{2}v^{7}z^{6}− 210uv^{5}z^{4} − 350u^{3}v^{3}z^{4} + 154u^{4}v^{3}z^{5} − 28u^{5}v^{3}z^{6} + 80u^{3}v^{3}z^{3} − 70u^{4}v^{3}z^{4} + 280u^{2}v^{3}z^{3} − 140u^{2}v^{3}z^{4} + 238u^{3}v^{3}z^{5}−140u^{4}v^{3}z^{6}+28v^{3}z^{5}u^{2}−56v^{3}z^{6}u^{3}+20u^{6}v^{3}z^{7}−10u^{7}v^{3}z^{8}+5u^{6}v^{2}z^{8}+20u^{5}v^{3}z^{7}+5u^{6}v^{3}z^{8}+ 40v^{3}z^{7}u^{4} − 10v^{3}z^{8}u^{5} − 10u^{7}z^{8}v^{2} + 392v^{4}u^{4}z^{6} − 100v^{4}u^{4}z^{7} − 100v^{4}u^{5}z^{7} + 20v^{4}z^{8}u^{5} + 120v^{5}u^{7}z^{8} + 5v^{4}u^{6}z^{8}−95v^{5}u^{6}z^{8}−20v^{5}z^{8}u^{5}+5v^{4}z^{8}u^{4}+20v^{4}u^{7}z^{8}−60v^{6}u^{3}z^{7}−15v^{5}u^{4}z^{8}+60v^{6}z^{8}u^{5}+380v^{6}u^{6}z^{7}+ 130v^{7}u^{7}z^{8}−75v^{6}u^{6}z^{8}−20v^{6}u^{7}z^{8}+15v^{6}z^{8}u^{4}+30v^{9}u^{7}z^{8}−80v^{8}u^{7}z^{8}−120u^{6}z^{2}+110u^{8}z^{2}v^{3}−60z^{3}u^{6}v+ 280v^{8}u^{8}z^{2} − 140v^{8}u^{7}z)

▷ A_{1}(z, u, v) = \frac{6}{35}(1 − z)^{3}(5z^{3}u^{3} + 5z^{2}u^{3} − 20u^{2}z^{2} + 3zu^{3} − 16u^{2}z + 28uz + u^{3} − 14 − 6u^{2} +

14u)(2uv − u − 1)u^{2}v^{4}

▷ A_{2}(z, u, v) = −\frac{6}{35}v^{4}(u − 2v + 1)(1 − uz)^{3}(−5z^{3}u^{3} + 20z^{2}u^{3} − 5u^{2}z^{2} − 28zu^{3} + 16u^{2}z − 3uz +

14u^{3} − 14u^{2} + 6u − 1)

▷ A_{3}(z, u, v) = \frac{1}{35}(−672v^{7}u^{6}z^{6}−3444u^{4}v^{4}z+720v^{8}u^{8}z^{7}−600v^{7}u^{8}z^{7}+1848v^{7}u^{5}z^{5}−2520v^{4}u^{6}z−

1008v^{8}u^{8}z^{6} +1260v^{5}u^{3}z +504u^{5}v^{8}z^{5} −6888v^{4}u^{5}z −240v^{7}u^{7}z^{7} +2016u^{7}v^{7}z^{6} +630u^{4}v^{8} +4704u^{5}v^{4} + 5040v^{6}u^{4} − 3564u^{5}v^{3} + 5166u^{4}v^{4} − 2394u^{4}v^{3} + 1848v^{4}u^{3} + 468v^{2}u^{4} − 180v^{8}u^{8}z^{8} − 252u^{3}v^{4}z − 420u^{8}v^{7}z^{5}+1512v^{5}u^{6}z+6552v^{5}u^{5}z+6048v^{5}u^{4}z−1680v^{7}u^{5}z^{4}+150v^{7}u^{8}z^{8}+840u^{8}v^{7}z^{6}+504u^{8}v^{8}z^{5}− 3528v^{7}u^{7}z^{5} + 84v^{10}z^{5}u^{7} + 1680v^{9}u^{7}z^{6} + 336v^{9}u^{8}z^{6} + 6552v^{8}z^{5}u^{6} − 168v^{10}u^{7}z^{6} + 2520v^{9}u^{6}z^{6} − 720v^{9}u^{7}z^{7}+1080v^{8}u^{7}z^{7}−2016v^{8}u^{6}z^{6}+840v^{10}u^{6}z^{5}−1848v^{9}u^{7}z^{5}+60v^{9}u^{8}z^{8}−3528v^{8}u^{7}z^{6}−168v^{9}u^{8}z^{5}− 4620v^{9}u^{6}z^{5}−3864v^{9}u^{5}z^{5}−5040v^{9}u^{5}z^{3}−240v^{9}u^{8}z^{7}+60u^{6}−420u^{5}v^{7}−378v^{3}u^{3}+3780v^{6}u^{3}+126v^{4}u^{2}− 4032u^{5}v^{5} −3654v^{5}u^{3} −3150u^{4}v^{7} +2520u^{5}v^{6} +1050u^{6}v^{4} −630u^{6}v^{5} −2310v^{7}u^{3} −630v^{5}u^{2} −6300v^{5}u^{4} + 1062v^{2}u^{6} − 1332v^{3}u^{6} + 1512v^{2}u^{5} + 630u^{2}v^{8} + 1260v^{6}u^{2} − 1260v^{7}u^{2} − 270u^{5}v − 420u^{6}v + 1260v^{8}u^{3} + 126v^{5}u^{7} − 252u^{7}v^{4} − 162u^{7}v^{2} + 756v^{3}u^{4}z + 4032v^{3}u^{5}z − 5040u^{5}v^{6}z − 5040v^{6}u^{4}z + 4200v^{7}u^{5}z^{2} + 840v^{7}u^{5}z + 3360v^{7}u^{4}z + 4788v^{8}u^{7}z^{5} − 1512v^{7}u^{6}z^{5} + 1680v^{9}u^{5}z^{2} − 1680v^{9}u^{6}z^{3} + 5040v^{8}u^{6}z^{3} − 4200u^{6}v^{7}z^{3} + 2100v^{7}u^{7}z^{4} − 5040v^{8}u^{5}z^{2} + 6300v^{9}u^{5}z^{4} + 4200v^{9}u^{6}z^{4} − 5040v^{8}u^{5}z^{4} − 1680v^{7}u^{5}z^{3} + 7560v^{8}u^{5}z^{3} + 840v^{9}u^{7}z^{4} + 5040v^{7}u^{6}z^{4} − 2520v^{8}u^{7}z^{4} − 420v^{10}u^{6}z^{4} − 8820v^{8}u^{6}z^{4} + 1008u^{5}v^{8}z^{6} + 2520u^{4}v^{8}z^{4}−2016u^{4}v^{8}z^{5}+2520v^{7}u^{3}z+360v^{7}u^{6}z^{7}−360v^{8}u^{5}z^{7}+840v^{10}u^{4}z^{5}−168v^{9}u^{3}z^{6}−336v^{9}u^{4}z^{6}+ 252v^{7}u^{4}z^{5} −252v^{8}u^{3}z^{5} −168v^{10}u^{4}z^{6} −30v^{9}u^{5}z^{8} +84v^{10}u^{3}z^{5} −336v^{9}u^{4}z^{5} +672v^{9}u^{3}z^{5} +120v^{9}u^{4}z^{7} + 84v^{9}u^{2}z^{5} + 30v^{9}u^{6}z^{8} + 504v^{8}u^{4}z^{6} + 1260v^{8}u^{3}z^{4} + 90v^{8}u^{6}z^{8} − 504v^{7}u^{5}z^{6} − 420v^{9}u^{2}z^{4} − 840v^{9}u^{3}z^{4} + 540u^{6}vz+3096u^{6}v^{3}z−2088u^{6}v^{2}z−1260v^{7}u^{4}z^{4}−600v^{9}u^{6}z^{7}+1008v^{9}u^{5}z^{6}−420v^{10}z^{4}u^{3}−1680v^{10}u^{4}z^{4}− 672v^{10}z^{6}u^{5} + 840v^{10}u^{3}z^{3} + 2520v^{9}u^{4}z^{4} + 1512v^{10}u^{5}z^{5} − 2520v^{8}u^{3}z^{3} + 840v^{9}u^{2}z^{3} − 840v^{9}u^{2}z^{2} + 420v^{9}u^{2}z+120v^{10}z^{7}u^{5}−936u^{5}v^{2}z−4200v^{9}u^{4}z^{3}+2520u^{3}v^{8}z^{2}+840u^{3}v^{9}z^{2}−2520v^{6}u^{3}z+2520v^{7}u^{4}z^{3}+ 3360u^{4}v^{9}z^{2} − 2520u^{3}v^{8}z + 1260v^{8}u^{4}z − 2520v^{8}u^{4}z^{2} − 840v^{9}u^{4}z − 420v^{9}u^{3}z − 252v^{5}u^{7}z + 420u^{7}v^{4}z − 432u^{7}v^{3}z − 36u^{7}v^{2}z − 2520v^{7}u^{4}z^{2} + 342u^{7}v^{3} − 210v^{9}u^{3} − 30u^{8} + 30u^{7} + 120u^{7}v^{10}z^{7} − 840u^{4}v^{10}z^{2} + 420u^{3}v^{10}z+840u^{5}v^{10}z^{3}+1680u^{4}v^{10}z^{3}−840u^{3}v^{10}z^{2}+360u^{8}v^{2}z+300u^{7}vz−1680u^{5}v^{10}z^{4}−252u^{8}v^{3}z− 240u^{8}vz −210u^{2}v^{9} −672u^{6}v^{10}z^{6} −30u^{6}v^{10}z^{8} −30u^{7}v^{10}z^{8} +240u^{6}v^{10}z^{7} +84u^{8}v^{4}z −30u^{7}v +120u^{8}v − 90v^{7}u^{7}z^{8} + 120v^{9}u^{7}z^{8} − 90v^{8}u^{7}z^{8} − 120u^{7}z + 60u^{8}z + 126u^{8}v^{3} − 180u^{8}v^{2} − 42u^{8}v^{4})

▷ A_{4}(z, u, v) = \frac{1}{35}(30 − 840v^{7}u^{6}z^{6} + 6888u^{4}v^{4}z − 1080v^{8}u^{8}z^{7} − 252u^{9}v^{7}z^{5} − 1008u^{9}v^{8}z^{6} +

240v^{7}u^{8}z^{7}−1260u^{9}v^{8}z^{4}+420v^{7}u^{5}z^{5}+1260v^{7}u^{8}z^{4}+252v^{4}u^{6}z+2016v^{8}u^{8}z^{6}−1512v^{5}u^{3}z+360u^{10}v^{8}z^{7}− 504u^{5}v^{8}z^{5} − 3096u^{3}v^{3}z − 360u^{9}v^{7}z^{7} + 3444v^{4}u^{5}z − 300vu^{2}z + 600v^{7}u^{7}z^{7} + 504v^{7}u^{9}z^{6} + 840v^{9}u^{5}z − 420v^{4}u^{2}z − 2016u^{7}v^{7}z^{6} + 420u^{2}v + 30uv − 60uz + 1260u^{6}v^{7} − 630u^{4}v^{8} − 630u^{6}v^{8} − 1260u^{5}v^{8} − 1260u^{6}v^{6} +210v^{9}u^{5} −1512v^{2}u^{3} −1848u^{5}v^{4} −5040v^{6}u^{4} +378u^{5}v^{3} −5166u^{4}v^{4} +2394u^{4}v^{3} −4704v^{4}u^{3} − 468v^{2}u^{4} − 360uv^{2}z + 240uvz + 2520v^{7}u^{6}z^{2} + 180v^{8}u^{8}z^{8} + 2520u^{6}v^{8}z^{2} + 2520u^{3}v^{4}z − 1848u^{8}v^{7}z^{5} + 252v^{5}u^{2}z − 1260v^{5}u^{6}z − 6048v^{5}u^{5}z − 6552v^{5}u^{4}z − 420v^{9}u^{7}z − 2100v^{7}u^{5}z^{4} − 150v^{7}u^{8}z^{8} + 90v^{8}u^{9}z^{8} + 90u^{9}v^{7}z^{8}−90u^{10}v^{8}z^{8}+672u^{8}v^{7}z^{6}−504u^{8}v^{8}z^{5}+1512v^{7}u^{7}z^{5}+1680v^{10}u^{8}z^{4}+840v^{9}u^{9}z^{4}−840v^{10}z^{5}u^{7}− 504v^{8}u^{10}z^{6} − 1680v^{9}u^{7}z^{6} − 2520v^{9}u^{8}z^{6} − 4788v^{8}z^{5}u^{6} + 2016u^{9}v^{8}z^{5} − 240v^{10}u^{9}z^{7} + 672v^{10}u^{8}z^{6} + 168v^{10}u^{7}z^{6}+420v^{10}u^{9}z^{4}−840v^{10}u^{9}z^{5}−336v^{9}u^{6}z^{6}+240v^{9}u^{7}z^{7}+420v^{9}u^{10}z^{4}−720v^{8}u^{7}z^{7}+1008v^{8}u^{6}z^{6}− 84v^{10}u^{6}z^{5}+4620v^{9}u^{7}z^{5}−60v^{9}u^{8}z^{8}+3528v^{8}u^{7}z^{6}−120v^{10}u^{8}z^{7}+3864v^{9}u^{8}z^{5}−840v^{10}u^{8}z^{3}+840v^{10}u^{7}z^{2}− 420v^{10}u^{6}z+1848v^{9}u^{6}z^{5}+168v^{9}u^{5}z^{5}+30v^{10}u^{9}z^{8}−1512v^{10}u^{8}z^{5}+840v^{9}u^{8}z^{2}+1680v^{9}u^{5}z^{3}+720v^{9}u^{8}z^{7}− 840v^{9}u^{9}z^{3}+30v^{9}u^{11}z^{8}−30v^{9}u^{10}z^{8}+30v^{10}u^{10}z^{8}+600v^{9}u^{9}z^{7}+180v^{2}−120v+42v^{4}−126v^{3}+120u^{2}z− 60u^{2} − 30u + 2310u^{5}v^{7} − 342v^{3}u + 3564v^{3}u^{3} + 162v^{2}u − 1062u^{2}v^{2} − 2520v^{6}u^{3} + 210v^{9}u^{6} + 270u^{3}v − 1050v^{4}u^{2} + 252v^{4}u + 1332v^{3}u^{2} + 3654u^{5}v^{5} − 126v^{5}u + 4032v^{5}u^{3} + 3150u^{4}v^{7} − 3780u^{5}v^{6} − 126u^{6}v^{4} + 630u^{6}v^{5} + 420v^{7}u^{3} + 630v^{5}u^{2} + 6300v^{5}u^{4} + 672v^{10}u^{9}z^{6} + 336v^{9}u^{9}z^{5} − 84v^{10}u^{10}z^{5} − 120v^{9}u^{11}z^{7} + 252v^{8}u^{10}z^{5} − 672v^{9}u^{10}z^{5} + 168v^{9}u^{11}z^{6} − 84v^{9}u^{11}z^{5} − 1008v^{9}u^{9}z^{6} − 120v^{10}u^{10}z^{7} − 120v^{9}u^{9}z^{8} + 168v^{10}u^{10}z^{6} + 336v^{9}u^{10}z^{6} − 4032v^{3}u^{4}z − 756v^{3}u^{5}z − 1680v^{10}u^{7}z^{3} + 840v^{10}u^{6}z^{2} + 5040u^{5}v^{6}z + 2520u^{6}v^{8}z+5040v^{6}u^{4}z−4200v^{7}u^{5}z^{2}−3360v^{7}u^{5}z−840v^{9}u^{7}z^{2}−840v^{7}u^{4}z−6552v^{8}u^{7}z^{5}−1260v^{8}u^{5}z+ 936v^{2}u^{4}z + 3528v^{7}u^{6}z^{5} − 1680v^{9}u^{5}z^{2} − 840v^{10}u^{6}z^{3} + 5040v^{9}u^{6}z^{3} − 7560v^{8}u^{6}z^{3} − 2520v^{8}u^{7}z^{2} + 1680u^{6}v^{7}z^{3} + 1680v^{7}u^{7}z^{4} − 2520v^{8}u^{8}z^{4} + 5040v^{8}u^{5}z^{2} − 840v^{9}u^{5}z^{4} − 4200v^{9}u^{6}z^{4} + 2520v^{8}u^{5}z^{4} + 420v^{9}u^{6}z + 432u^{2}v^{3}z + 36u^{2}v^{2}z + 4200v^{7}u^{5}z^{3} + 4200v^{9}u^{7}z^{3} − 540vzu^{3} + 2088v^{2}zu^{3} − 84v^{4}zu + 252v^{3}uz − 5040v^{8}u^{5}z^{3} + 2520v^{8}u^{8}z^{3} − 2520v^{9}u^{8}z^{4} − 6300v^{9}u^{7}z^{4} + 1680v^{10}u^{7}z^{4} − 5040v^{7}u^{6}z^{4} + 5040v^{8}u^{7}z^{4} + 420v^{10}u^{6}z^{4} + 8820v^{8}u^{6}z^{4} − 3360v^{9}u^{6}z^{2} − 2520v^{7}u^{7}z^{3} − 2520v^{7}u^{6}z + 2520v^{6}u^{6}z)

A.2. The polynomials B_{i}(z, u).

▷ B_{0}(z, u) = 10uz−120u^{2}z+1140u^{2}z^{2}−8190u^{5}z^{2}−4236z^{5}u^{6}−1980z^{3}u^{5}−680z^{4}u^{6}−980u^{4}z− 605z^{4}u^{4} −4826z^{5}u^{5} +4354z^{6}u^{6} −5020z^{3}u^{4} +8585z^{4}u^{5} +500zu^{3} +2980z^{3}u^{3} −4140z^{2}u^{3} +8790z^{2}u^{4} − 1470u^{6} − 490u^{2} + 1470u^{3} − 2450u^{4} + 70u + 980u^{6}z + 2450u^{5} + 8190u^{6}z^{2} + 1040u^{9}z^{7} + 960u^{10}z^{7} − 1040u^{7}z^{7} +518u^{7}z^{6} +490u^{7} −70u^{8} +4236u^{8}z^{5} +605u^{9}z^{4} −518u^{8}z^{6} −4354u^{9}z^{6} −55u^{8}z^{8} −8585u^{8}z^{4} + 4826u^{9}z^{5} + 112u^{10}z^{6} − 500u^{7}z + 55u^{9}z^{8} − 390u^{10}z^{8} + 490u^{10}z^{4} + 5020u^{8}z^{3} − 8790u^{7}z^{2} + 4140u^{8}z^{2} − 2980u^{9}z^{3} +150u^{11}z^{8} +680u^{7}z^{4} −1694u^{10}z^{5} −440u^{11}z^{7} +196u^{11}z^{6} +462u^{11}z^{5} +1980u^{7}z^{3} −112u^{5}z^{6} − 140uz^{2} + 14u^{2}z^{5} + 1694u^{4}z^{5} + 40u^{5}z^{8} − 960u^{6}z^{7} + 440u^{5}z^{7} − 5u^{4}z^{8} − 28u^{2}z^{6} + 140u^{3}z^{6} + 20u^{3}z^{7} − 70uz^{4} + 14z^{5}u − 140u^{4}z^{7} − 196u^{4}z^{6} − 150u^{6}z^{8} − 462u^{3}z^{5} + 350u^{2}z^{4} − 490u^{3}z^{4} + 140uz^{3} − 980u^{2}z^{3} − 20u^{13}z^{7} + 140u^{10}z^{2} − 140u^{11}z^{3} + 70u^{12}z^{4} + 5u^{13}z^{8} + 28u^{13}z^{6} − 14u^{13}z^{5} − 40u^{12}z^{8} + 140u^{12}z^{7} − 140u^{12}z^{6} − 350u^{11}z^{4} − 14u^{12}z^{5} − 10u^{9}z + 120u^{8}z + 390u^{7}z^{8} − 1140u^{9}z^{2} + 980u^{10}z^{3}

▷ B_{1}(z, u) = −6(1 + u)u^{5}(1 − z)^{3}(5z^{3}u^{3} + 5z^{2}u^{3} − 20u^{2}z^{2} + 3u^{3}z − 16u^{2}z + 28uz + u^{3} − 6u^{2} +

14u − 14)

▷ B_{2}(z, u) = 6(1 + u)(5z^{3}u^{3} − 20z^{2}u^{3} + 5u^{2}z^{2} − 16u^{2}z + 3uz + 28u^{3}z − 6u − 14u^{3} + 14u^{2} +

1)(1 − uz)^{3}

A.3. The polynomials C_{i}(z, u).

▷ C_{0}(z, u) = 950uz −6360u^{2}z +5120u^{2}z^{2} +10850u^{5}z^{2} −58440z^{5}u^{6} −39420z^{3}u^{5} +32210z^{4}u^{6} − 22540u^{4}z + 275z^{4}u^{4} − 38218z^{5}u^{5} + 38066z^{6}u^{6} − 32540z^{3}u^{4} + 69865z^{4}u^{5} + 17260zu^{3} + 17660z^{3}u^{3} − 16280z^{2}u^{3} +29070z^{2}u^{4} +22540u^{6}z −10850u^{6}z^{2} +13960u^{9}z^{7} +7320u^{10}z^{7} −13960u^{7}z^{7} +21574u^{7}z^{6} + 58440u^{8}z^{5} − 275u^{9}z^{4} − 21574u^{8}z^{6} − 38066u^{9}z^{6} + 1600u^{8}z^{8} − 69865u^{8}z^{4} + 38218u^{9}z^{5} − 1540u^{10}z^{6} − 17260u^{7}z − 1600u^{9}z^{8} − 2715u^{10}z^{8} + 3710u^{10}z^{4} + 32540u^{8}z^{3} − 29070u^{7}z^{2} + 16280u^{8}z^{2} − 17660u^{9}z^{3} + 885u^{11}z^{8} − 32210u^{7}z^{4} − 9352u^{10}z^{5} − 2680u^{11}z^{7} + 1484u^{11}z^{6} + 2310u^{11}z^{5} + 39420u^{7}z^{3} + 1540u^{5}z^{6} − 700uz^{2}+28u^{2}z^{5}+9352u^{4}z^{5}+215u^{5}z^{8}−7320u^{6}z^{7}+2680u^{5}z^{7}−25u^{4}z^{8}−140u^{2}z^{6}+784u^{3}z^{6}+100u^{3}z^{7}− 350uz^{4}+70z^{5}u−760u^{4}z^{7}−1484u^{4}z^{6}−885u^{6}z^{8}−2310u^{3}z^{5}+1960u^{2}z^{4}−3710u^{3}z^{4}+700uz^{3}−5320u^{2}z^{3}− 100u^{13}z^{7} + 700u^{10}z^{2} − 700u^{11}z^{3} + 350u^{12}z^{4} + 25u^{13}z^{8} + 140u^{13}z^{6} − 70u^{13}z^{5} − 215u^{12}z^{8} + 760u^{12}z^{7} − 784u^{12}z^{6} − 1960u^{11}z^{4} − 28u^{12}z^{5} − 950u^{9}z + 6360u^{8}z + 2715u^{7}z^{8} − 5120u^{9}z^{2} + 5320u^{10}z^{3}

▷ C_{1}(z, u) = 18480u^{5}z^{2} − 36960u^{6}z^{3} − 18480z^{5}u^{6} − 18480z^{3}u^{5} + 36960z^{4}u^{6} + 3024u^{4}z − 1848z^{5}u^{5}+3696z^{6}u^{6}+9240z^{4}u^{5}−504zu^{3}−1260u^{6}+252u^{2}−1260u^{3}+2268u^{4}+6384u^{6}z−16800u^{5}z+ 588u^{5} + 18480u^{6}z^{2} − 2640u^{9}z^{7} − 2640u^{7}z^{7} + 14784u^{7}z^{6} + 1212u^{7} − 600u^{8} + 120u^{9} − 18480u^{8}z^{5} + 14784u^{8}z^{6}+3696u^{9}z^{6}+660u^{8}z^{8}+9240u^{8}z^{4}−1848u^{9}z^{5}−3864u^{7}z+660u^{9}z^{8}+36960u^{7}z^{4}−5280u^{8}z^{7}− 18480u^{7}z^{3} − 33264u^{7}z^{5} − 240u^{9}z + 1440u^{8}z

▷ C_{2}(z, u) = −120 + 240uz − 1440u^{2}z − 18480u^{5}z^{2} + 36960u^{6}z^{3} + 18480z^{5}u^{6} + 18480z^{3}u^{5} − 36960z^{4}u^{6} − 6384u^{4}z + 1848z^{5}u^{5} − 3696z^{6}u^{6} − 9240z^{4}u^{5} + 3864zu^{3} + 1260u^{6} − 1212u^{2} + 1260u^{3} − 588u^{4} + 600u − 3024u^{6}z + 16800u^{5}z − 2268u^{5} − 18480u^{6}z^{2} + 2640u^{9}z^{7} + 2640u^{7}z^{7} − 14784u^{7}z^{6} − 252u^{7} + 18480u^{8}z^{5} − 14784u^{8}z^{6} − 3696u^{9}z^{6} − 660u^{8}z^{8} − 9240u^{8}z^{4} + 1848u^{9}z^{5} + 504u^{7}z − 660u^{9}z^{8} − 36960u^{7}z^{4} + 5280u^{8}z^{7} + 18480u^{7}z^{3} + 33264u^{7}z^{5}

A.4. The polynomials D_{i}(z, u).

▷ D_{0}(z, u) = −\frac{u}{350}(−1960uz^{3} −19000u^{3}z^{3} +19050u^{3}z^{2} −84uz^{5} +370z +840uz^{4} +4844u^{3}z^{5} +

400u^{3}z^{4} + 2520u^{9}z^{4} + 1960u^{9}z^{3} + 140u^{9}z^{2} − 8900u^{8}z^{3} − 1650u^{8}z^{2} + 336u^{2}z^{6} − 644u^{2}z^{5} − 2520u^{2}z^{4} + 8900u^{2}z^{3} − 7650u^{2}z^{2} − 28uz^{6} + 2450u − 1008u^{3}z^{6} + 7650u^{7}z^{2} + 1650uz^{2} + 20u^{2}z^{7} − 350 − 7350u^{2} + 7350u^{5} − 2450u^{6} − 140z^{2} − 70z^{4} + 140z^{3} + 14z^{5} − 2340uz + 5900u^{2}z − 6860u^{3}z + 6860u^{5}z − 5900u^{6}z + 2340u^{7}z − 370u^{8}z − 20790u^{4}z^{2} + 5840u^{4}z^{3} + 22630u^{4}z^{4} − 15546u^{4}z^{5} + 20790u^{5}z^{2} − 17040u^{5}z^{4} − 1528u^{5}z^{5} − 19050u^{6}z^{2} − 5840u^{6}z^{3} + 17040u^{6}z^{4} − 400u^{8}z^{4} + 15546u^{8}z^{5} − 4844u^{9}z^{5} + 12250u^{3} − 12250u^{4} + 350u^{7} + 19000u^{7}z^{3} − 22630u^{7}z^{4} + 1)

▷ D_{1}(z, u) = \frac{−6u}{35}(−7+42u−98u^{2} −98u^{5} +82u^{6} +14uz −98u^{2}z +294u^{3}z −630u^{4}z +434u^{5}z −

238u^{6}z + 74u^{7}z − 10u^{8}z + 280u^{4}z^{2} − 280u^{4}z^{3} + 140u^{4}z^{4} − 28u^{4}z^{5} + 280u^{5}z^{2} − 560u^{5}z^{3} + 560u^{5}z^{4} − 280u^{5}z^{5} − 280u^{6}z^{3} + 560u^{6}z^{4} − 504u^{6}z^{5} − 28u^{8}z^{5} + 98u^{3} + 28u^{4} − 32u^{7} + 5u^{8} + 140u^{7}z^{4} − 280u^{7}z^{5} + 224u^{7}z^{6} + 224u^{6}z^{6} + 56u^{8}z^{6} − 80u^{7}z^{7} − 40u^{8}z^{7} + 10u^{8}z^{8} + 56u^{5}z^{6} − 40u^{6}z^{7} + 10u^{7}z^{8})

▷ D_{2}(z, u) = −\frac{1}{35}(30 − 192u + 492u^{2} + 588u^{5} − 588u^{6} − 60uz + 444u^{2}z − 1428u^{3}z + 2604u^{4}z −

3780u^{5}z + 1764u^{6}z − 588u^{7}z + 84u^{8}z + 1680u^{5}z^{2} − 1680u^{5}z^{3} + 840u^{5}z^{4} − 168u^{5}z^{5} + 1680u^{6}z^{2} − 3360u^{6}z^{3} + 3360u^{6}z^{4} − 1680u^{6}z^{5} + 840u^{8}z^{4} − 1680u^{8}z^{5} − 168u^{9}z^{5} − 588u^{3} + 168u^{4} + 252u^{7} − 42u^{8} − 1680u^{7}z^{3} + 3360u^{7}z^{4} − 3024u^{7}z^{5} + 1344u^{7}z^{6} + 336u^{6}z^{6} + 1344u^{8}z^{6} + 336u^{9}z^{6} − 240u^{7}z^{7} − 240u^{9}z^{7} − 480u^{8}z^{7} + 60u^{8}z^{8} + 60u^{9}z^{8})

Appendix B. Tables

The next Tables collect the full statements of the theorems, considering the general cases as they appear in the main text of the paper as well as the special cases not listed there.

j  d_{n,j}

\frac{6}{5}, for n ≥ 5

1  d_{1,1} = 1, d_{2,1} = \frac{3}{2}, d_{3,1} = 1, d_{4,1} = \frac{5}{4}

\frac{18}{5} − \frac{3}{n}, for n ≥ 6

2  d_{2,2} = \frac{3}{2}, d_{3,2} = 3, d_{4,2} = \frac{11}{4}, d_{5,2} = 3

\frac{136}{35} − \frac{6}{n}, for n ≥ 7

3  d_{3,3} = 1, d_{4,3} = \frac{11}{4}, d_{5,3} = \frac{13}{5}, d_{6,3} = \frac{29}{10}

\frac{9}{2} − \frac{9}{n}, for n ≥ 8

4  d_{4,4} = \frac{5}{4}, d_{5,4} = 3, d_{6,4} = \frac{29}{10}, d_{7,4} = \frac{113}{35}

5 ≤ j ≤ n − 4  \begin{matrix}−\frac{12}{7}H_{n} + \frac{12}{7}H_{j} + \frac{12}{7}H_{n+1−j} \\ −\frac{6}{7j} − \frac{6}{7(n + 1 − j)} + \frac{79}{70} \\ −\frac{3(3j − 5)}{7n} + \frac{6(j − 1)^{2}}{7n\underline{2}} + \frac{2(2j − 3)(j − 1)\underline{2}}{7n\underline{3}} \\ +\frac{3(j − 2)(j − 1)\underline{3}}{7n\underline{4}} − \frac{3(2j − 5)(j − 1)\underline{4}}{7n\underline{5}} + \frac{2(j − 3)(j − 1)\underline{5}}{7n\underline{6}} \end{matrix}

Table 1. Average number of descendants of the j^{th} node in a random LBST of size n.

j  d_{n,j}^{(2)}

\frac{2}{5}  for n ≥ 5

1  d_{1,1}^{(2)} = 0, d_{2,1}^{(2)} = 1, d_{3,1}^{(2)} = 0, d_{4,1}^{(2)} = \frac{1}{2}

\frac{36}{5}H_{n} − \frac{206}{25} for n ≥ 6

2  d_{2,2}^{(2)} = 1, d_{3,2}^{(2)} = 6, d_{4,2}^{(2)} = \frac{13}{2}, d_{5,2}^{(2)} = \frac{41}{5}

\frac{72}{5}H_{n} − \frac{4534}{175} + \frac{6}{n} for n ≥ 7

3  d_{3,3}^{(2)} = 0, d_{4,3}^{(2)} = \frac{13}{2}, d_{5,3}^{(2)} = 8, d_{6,3}^{(2)} = \frac{52}{5}

\frac{108}{5}H_{n} − \frac{7886}{175} − \frac{36}{5(n − 1)} + \frac{126}{5n} for n ≥ 8

4  d_{4,4}^{(2)} = \frac{1}{2}, d_{5,4}^{(2)} = \frac{41}{5}, d_{6,4}^{(2)} = \frac{52}{5}, d_{7,4}^{(2)} = \frac{468}{35}

5 ≤ j ≤ n − 4  \begin{matrix}(\frac{36n}{5} − \frac{12}{35}) H_{n} + (\frac{36j}{5} − \frac{36n}{5} − \frac{48}{7}) H_{n+1−j} + (\frac{12}{35} − \frac{36j}{5} ) H_{j} \\ −\frac{132}{35j} − \frac{132}{35(n + 1 − j)} + \frac{3489}{175} − \frac{33j}{5} + (\frac{66}{7} − \frac{429j}{35} + \frac{33j^{2}}{5} ) \frac{1}{n} \\ +\frac{132(j − 1)^{2}}{35n\underline{2}} + \frac{44(2j − 3)(j − 1)\underline{2}}{35n\underline{3}} + \frac{66(j − 2)(j − 1)\underline{3}}{35n\underline{4}} \\ −\frac{66(2j − 5)(j − 1)\underline{4}}{35n\underline{5}} + \frac{44(j − 3)(j − 1)\underline{5}}{35n\underline{6}} \end{matrix}

Table 2. Second factorial moment of the number of descendants of the j^{th} node in a random LBST of size n.

m  p_{n,m} = ℙ [D_{n} = m]

\frac{3}{7} (1 + \frac{1}{n})  for n ≥ 6

1  p_{1,1} = 1, p_{2,1} = \frac{1}{4}, p_{3,1} = \frac{2}{9}, p_{4,1} = \frac{1}{8}, p_{5,1} = \frac{13}{125}

\frac{1}{7} (1 + \frac{1}{n}) , for n ≥ 6

2  p_{2,2} = \frac{1}{4}, p_{3,2} = 0, p_{4,2} = \frac{1}{16}, p_{5,2} = \frac{4}{125}

\frac{3}{35} (1 + \frac{1}{n}) , for n ≥ 6

3  p_{3,3} = \frac{1}{9}, p_{4,3} = 0, p_{5,3} = \frac{3}{5}

\frac{2}{35} (1 + \frac{1}{n})  for n ≥ 6

4  p_{4,4} = \frac{1}{16}, p_{5,4} = 0

5 ≤ m < n  \frac{12}{7} \frac{(n + 2 + m)(n − 1 − m)}{n^{2}(m + 2)(m + 1)} − \frac{12}{7} \frac{m\underline{5}}{n^{2}n\underline{5}} + \frac{12}{7n^{2}}

n  \frac{1}{n}

Table 3. Probability that a random node in a random LBST of size n has m descendants.

j  a_{n,j}

\frac{6}{5}H_{n} − \frac{6}{25}, for n ≥ 5

1  a_{1,1} = 1, a_{2,1} = \frac{3}{2}, a_{3,1} = 2, a_{4,1} = \frac{9}{4}

\frac{6}{5}H_{n} − \frac{21}{25}, for n ≥ 6

2  a_{2,2} = \frac{3}{2}, a_{3,2} = 1, a_{4,2} = \frac{7}{4}, a_{5,2} = \frac{19}{10}

\frac{6}{5}H_{n} − \frac{87}{175} − \frac{6}{5} \frac{1}{n\underline{2}} , for n ≥ 7

3  a_{3,3} = 2, a_{4,3} = \frac{7}{4}, a_{5,3} = \frac{11}{5}, a_{6,3} = \frac{12}{5}

\frac{6}{5}H_{n} − \frac{11}{25} − \frac{18}{5} \frac{1}{n\underline{2}} − \frac{12}{5} \frac{1}{n\underline{3}} , for n ≥ 8

4  a_{4,4} = \frac{9}{4}, a_{5,4} = \frac{19}{10}, a_{6,4} = \frac{12}{5}, a_{7,4} = \frac{18}{7}

5 ≤ j ≤ n − 4  \begin{matrix}\frac{24}{35}H_{n} + \frac{18}{35}H_{j} + \frac{18}{35}H_{n+1−j} \\ +\frac{12}{35j} + \frac{12}{35(n + 1 − j)} − \frac{279}{175} − \frac{6}{7n} \\ +\frac{18j}{35n} − \frac{12(j − 1)^{2}}{35n\underline{2}} − \frac{4(2j − 3)(j − 1)\underline{2}}{35n\underline{3}} \\ −\frac{6(j − 2)(j − 1)\underline{3}}{35n\underline{4}} + \frac{6(2j − 5)(j − 1)\underline{4}}{35n\underline{5}} − \frac{4(j − 3)(j − 1)\underline{5}}{35n\underline{6}} \end{matrix}

Table 4. Average number of ascendants of the j^{th} node in a random LBST of size n.

j  a_{n,j}^{(2)}

\frac{36}{25}H_{n}^{2} − \frac{144}{125}H_{n} − \frac{36}{25}H_{n}^{(2)} + \frac{766}{625} + \frac{3}{125} \frac{1}{n} − \frac{12}{125} \frac{1}{n − 1} + \frac{18}{125} \frac{1}{n − 2} − \frac{12}{125} \frac{1}{n − 3}

1  +\frac{3}{125} \frac{1}{n − 4}  for n ≥ 5

a_{1,1}^{(2)} = 0 , a_{2,1}^{(2)} = 1 , a_{3,1}^{(2)} = 2 , a_{4,1}^{(2)} = 3

\frac{36}{25}H_{n}^{2} − \frac{324}{125}H_{n} − \frac{36}{25}H_{n}^{(2)} + \frac{1726}{625} − \frac{27}{125} \frac{1}{n} + \frac{108}{125} \frac{1}{n − 1} − \frac{162}{125} \frac{1}{n − 2}

2  +\frac{108}{125} \frac{1}{n − 3} − \frac{27}{125} \frac{1}{n − 4}  for n ≥ 6

a_{2,2}^{(2)} = 1 , a_{3,2}^{(2)} = 0 , a_{4,2}^{(2)} = 2 , a_{5,2}^{(2)} = \frac{11}{5}

\frac{36}{25}H_{n}^{2} − (\frac{1548}{875} + \frac{72}{25} \frac{1}{n(n − 1)}) H_{n} − \frac{36}{25}H_{n}^{(2)} + \frac{77094}{30625} + \frac{2169}{875} \frac{1}{n} − \frac{2892}{875} \frac{1}{n − 1}

+\frac{1476}{875} \frac{1}{n − 2} − \frac{1044}{875} \frac{1}{n − 3} + \frac{321}{875} \frac{1}{n − 4} − \frac{36}{875} \frac{1}{n − 5} + \frac{6}{875} \frac{1}{n − 6}

3  − \frac{72}{25} \frac{1}{n^{2}} + \frac{72}{25} \frac{1}{(n − 1)^{2}}  for n ≥ 7

a_{3,3}^{(2)} = 2 , a_{4,3}^{(2)} = 2 , a_{5,3}^{(2)} = \frac{18}{5} , a_{6,3}^{(2)} = \frac{22}{5}

\frac{36}{25}H_{n}^{2} + (−\frac{204}{125} + \frac{144}{25} \frac{1}{n} − \frac{72}{25} \frac{1}{n − 1} − \frac{72}{25} \frac{1}{n − 2}) H_{n} − \frac{36}{25}H_{n}^{(2)} + \frac{11077}{4375}

+ \frac{5517}{875} \frac{1}{n} − \frac{786}{125} \frac{1}{n − 1} − \frac{246}{125} \frac{1}{n − 2} + \frac{378}{125} \frac{1}{n − 3} − \frac{177}{125} \frac{1}{n − 4} + \frac{54}{125} \frac{1}{n − 5}

4  − \frac{12}{125} \frac{1}{n − 6} + \frac{6}{875} \frac{1}{n − 7} − \frac{144}{25} \frac{1}{n^{2}} + \frac{72}{25} \frac{1}{(n − 1)^{2}} + \frac{72}{25} \frac{1}{(n − 2)^{2}}  for n ≥ 8 ,

a_{4,4}^{(2)} = 3 , a_{5,4}^{(2)} = \frac{11}{5} , a_{6,4}^{(2)} = \frac{22}{5} , a_{7,4}^{(2)} = \frac{36}{7}

THE ELECTRONIC JOURNAL OF COMBINATORICS 5 (1998), #R20 (APPENDIX) 10

\begin{matrix}j & a_{n,j}^{(2)} \\ 5 ≤ j ≤ n − 4 & \begin{matrix}\frac{576}{245}H_{n}H_{n−j} + \frac{576}{245}H_{n}H_{j} − \frac{288}{245}H_{n}^{2} + \frac{324}{1225}H_{n−j}^{2} − \frac{1368}{1225}H_{n−j}H_{j} + \frac{324}{1225}H_{j}^{2} \\ + (\frac{72}{42875} \frac{227j+525}{j} + \frac{792}{245} \frac{1}{n+1−j} − \frac{2}{1225} \frac{2j^{6}−6j^{5}−55j^{4}+120j^{3}+953j^{2}−1014j+540}{n} \\ + \frac{4}{245} \frac{(j−1)^{2}(j^{4}−4j^{3}−11j^{2}+30j+72)}{n−1} − \frac{4}{245} \frac{(j−1)(j−2)(2j^{4}−12j^{3}+13j^{2}+15j−36)}{n−2} \\ + \frac{8}{245} \frac{(j−1)^{2}(j−2)^{2}(j−3)^{2}}{n−3} − \frac{2}{245} \frac{(j−1)(j−2)(j−3)(j−4)(2j^{2}−10j+15)}{n−4} \\ + \frac{4}{1225} \frac{(j−1)(j−2)(j−3)^{2}(j−4)(j−5)}{n−5})H_{n} \\ + ( − \frac{36}{42875} \frac{5697j+70}{j} − \frac{288}{245} \frac{1}{n+1−j} + \frac{1}{6125} \frac{20j^{6}−102j^{5}−445j^{4}+2040j^{3}+8165j^{2}+402j+360}{n} \\ − \frac{2}{1225} \frac{(j−1)(10j^{5}−71j^{4}+14j^{3}+557j^{2}−42j−1224)}{n−1} + \frac{2}{1225} \frac{(j−1)(j−2)(20j^{4}−162j^{3}+319j^{2}+129j−612)}{n−2} \\ − \frac{4}{1225} \frac{(j−1)(j−2)(j−3)(10j^{3}−81j^{2}+194j−102)}{n−3} + \frac{1}{1225} \frac{(j−1)(j−2)(j−3)(j−4)(20j^{2}−142j+255)}{n−4} \\ − \frac{2}{6125} \frac{(j−1)(j−2)(j−3)(j−4)(j−5)(10j−51)}{n−5})H_{n−j} \\ + ( − \frac{36}{42875} \frac{2267j+2030}{j} − \frac{288}{245} \frac{1}{n+1−j} + \frac{1}{6125} \frac{20j^{6}−18j^{5}−655j^{4}+360j^{3}+10895j^{2}−20682j+10440}{n} \\ − \frac{2}{1225} \frac{(j−1)(10j^{5}−29j^{4}−154j^{3}+263j^{2}+882j−216)}{n−1} + \frac{2}{1225} \frac{(j−1)(j−2)(20j^{4}−78j^{3}−59j^{2}+171j−108)}{n−2} \\ − \frac{4}{1225} \frac{(j−1)(j−2)(j−3)(10j^{3}−39j^{2}+26j−18)}{n−3} + \frac{1}{1225} \frac{(j−1)(j−2)(j−3)(j−4)(20j^{2}−58j+45)}{n−4} \\ − \frac{2}{6125} \frac{(j−1)(j−2)(j−3)(j−4)(j−5)(10j−9)}{n−5})H_{j} \\ + \frac{288}{245}H_{n}^{(2)} − \frac{3204}{1225}H_{n−j}^{(2)} − \frac{3204}{1225}H_{j}^{(2)} + \frac{12}{1225} \frac{1}{j+4} − \frac{24}{175} \frac{1}{j+3} + \frac{792}{1225} \frac{1}{j+2} − \frac{2088}{1225} \frac{1}{j+1} \\ − \frac{53148}{42875} \frac{1}{j} − \frac{1044}{875} \frac{1}{j−1} + \frac{396}{875} \frac{1}{j−2} − \frac{12}{125} \frac{1}{j−3} + \frac{6}{875} \frac{1}{j−4} + \frac{144}{175} \frac{1}{j^{2}} + \frac{6}{875} \frac{1}{n−3−j} − \frac{12}{125} \frac{1}{n−2−j} \\ + \frac{396}{875} \frac{1}{n−1−j} − \frac{1044}{875} \frac{1}{n−j} − \frac{24}{8575} \frac{1123j−420}{(n+1−j)j} − \frac{2088}{1225} \frac{1}{n+2−j} + \frac{792}{1225} \frac{1}{n+3−j} − \frac{24}{175} \frac{1}{n+4−j} \\ + \frac{12}{1225} \frac{1}{n+5−j} − \frac{792}{245} \frac{1}{(n+1−j)^{2}} + \frac{2297696}{214375} \\ − \frac{5522j^{7}−8166j^{6}−167395j^{5}+142845j^{4}+3249683j^{3}−1044579j^{2}−2445930j+1096200}{643125 nj} \\ + \frac{2(2761j^{7}−12366j^{6}−25787j^{5}+116883j^{4}+176746j^{3}−361269j^{2}+120672j+22680)}{128625 (n−1)j} \\ − \frac{2(5522j^{7}−41298j^{6}+82913j^{5}+11427j^{4}−226090j^{3}+256698j^{2}−49482j−22680)}{128625 (n−2)j} \\ + \frac{4(j−1)(j−3)(2761j^{5}−17888j^{4}+38198j^{3}−28798j^{2}−5718j+3780)}{128625 (n−3)j} \\ − \frac{5522j^{7}−74430j^{6}+407885j^{5}−1158375j^{4}+1755053j^{3}−1233465j^{2}+169920j+113400}{128625 (n−4)j} \\ + \frac{2(2761j^{7}−45498j^{6}+299185j^{5}−999255j^{4}+1757614j^{3}−1456887j^{2}+313560j+113400)}{643125 (n−5)j} \\ + \frac{2(2j^{6}−6j^{5}−55j^{4}+120j^{3}+953j^{2}−1014j+540)}{1225 n^{2}} − \frac{4(j−1)^{2}(j^{4}−4j^{3}−11j^{2}+30j+72)}{245 (n−1)^{2}} \\ + \frac{4(j−1)(j−2)(2j^{4}−12j^{3}+13j^{2}+15j−36)}{245 (n−2)^{2}} − \frac{8(j−1)^{2}(j−2)^{2}(j−3)^{2})}{245 (n−3)^{2}} \\ + \frac{2(j−1)(j−2)(j−3)(j−4)(2j^{2}−10j+15)}{245 (n−4)^{2}} − \frac{4(j−1)(j−2)(j−3)^{2}(j−4)(j−5)}{1225 (n−5)^{2}} \end{matrix} \end{matrix}

Table 5. Second factorial moment of the number of ascendants of the j^{th} node in a random LBST of size n.