A137388 - OEIS

%I #13 Sep 26 2024 03:46:17

%S -1,-1,0,-1,0,6,-1,0,15,20,-1,0,27,64,45,-1,0,42,140,175,84,-1,0,60,

%T 256,450,384,140,-1,0,81,420,945,1134,735,216,-1,0,105,640,1750,2688,

%U 2450,1280,315,-1,0,132,924,2970,5544,6468,4752,2079,440,-1,0,162,1280,4725,10368,14700,13824,8505,3200,594

%N Triangle t(n,m)= (m^2-1) * binomial(n,m) * (n+2)/(n+2-m) read by rows, 0<=m<=n.

%C Row sums are -1, -1, 5, 34, 135, 440, 1289, 3530, 9227, 23308, 57357, ... = 3 + n - 2^(n+2) + n^2*2^(n-1) + n*2^n.

%F t(n,m) = (m-1)*(m+1)*binomial(n,m)*binomial(n+2,m)/binomial(n+1,m).

%e Triangle begins:

%e   -1;

%e   -1, 0;

%e   -1, 0, 6;

%e   -1, 0, 15, 20;

%e   -1, 0, 27, 64, 45;

%e   -1, 0, 42, 140, 175, 84;

%e   -1, 0, 60, 256, 450, 384, 140;

%e   -1, 0, 81, 420, 945, 1134, 735, 216;

%e   -1, 0, 105, 640, 1750, 2688, 2450, 1280, 315;

%e   -1, 0, 132, 924, 2970, 5544, 6468, 4752, 2079, 440;

%e   -1, 0, 162, 1280, 4725, 10368, 14700, 13824, 8505, 3200, 594;

%p A137388 := proc(n,m)

%p         (m^2-1)*binomial(n,m)*(n+2)/(n+2-m) ;

%p end proc:

%p seq(seq(A137388(n,m),m=0..n),n=0..14) ; # _R. J. Mathar_, Nov 10 2011

%t a0 = Table[Table[(n - 1)*(n + 1)*Binomial[m, n]*Binomial[m + 2, n]/Binomial[m + 1, n], {n,0, m}], {m, 0, 10}]; Flatten[a0]

%K tabl,sign,easy

%O 0,6

%A _Roger L. Bagula_ and _Gary W. Adamson_, Apr 10 2008