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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.
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%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