%I #14 Dec 26 2023 11:59:09
%S -1,3,6,-20,12,210,40,360,-3024,180,840,55440,1008,3360,60480,
%T -1235520,6720,18144,151200,32432400,51840,120960,665280,19958400,
%U -980179200,453600,950400,3991680,51891840,33522128640,4435200,8553600,29652480,242161920,10897286400,-1279935820800
%N Triangle t(n,m) read by rows which contains in row n integer values of n! * binomial(n+m+1,m+1) / binomial(n-m-1,m+1) sorted along increasing m.
%C These are (rational) solutions x to the equation binomial(n+m+1,m+1) - x*binomial(n-m-1,m+1) = 0, post-multiplied by n!.
%C Row sums are -1, 3, -14, 222, -2624, 56460, -1170672, 32608464, -959382720, 33579416160, -1268753731200,...
%e -1; # n=0, m=0
%e 3; # n=1, m=1
%e 6, -20; # n=2, m=0,2
%e 12, 210; # n=3, m=0,3
%e 40, 360, -3024; # n=4, m=0,1,4
%e 180, 840, 55440; # n=5, m=0,1,5
%e 1008, 3360, 60480, -1235520; # n=6, m=0,1,2,6
%e 6720, 18144, 151200, 32432400; # n=7, m=0,1,2,7
%e 51840, 120960, 665280, 19958400, -980179200; # n=8, m=0,1,2,3,8
%e 453600, 950400, 3991680, 51891840, 33522128640; # n=9, m=0,1,2,3,9
%e 4435200, 8553600, 29652480, 242161920, 10897286400, -1279935820800; # n=10, m=0,1,2,3,4,10
%t t[n_, m_] = Binomial[n + (m + 1), (m + 1)] - x*Binomial[n - (m + 1), (m + 1)];
%t (* if the solution exists it is made part of the array: if not, it is deleted*)
%t Table[Flatten[Table[If[Solve[t[n, m] == 0, x] == {}, {}, n!
%t x /. Solve[t[n, m] == 0, x]], {m, 0, n}], 1], {n, 0, 10}];
%t Flatten[%]
%K sign,tabf
%O 0,2
%A _Roger L. Bagula_, Dec 08 2010