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A110180 Triangle of generalized central trinomial coefficients.

%I A110180 #9 Mar 06 2017 02:58:54
%S A110180 1,1,1,1,1,1,1,3,1,1,1,7,5,1,1,1,19,13,7,1,1,1,51,49,19,9,1,1,1,141,
%T A110180 161,91,25,11,1,1,1,393,581,331,145,31,13,1,1,1,1107,2045,1441,561,
%U A110180 211,37,15,1,1,1,3139,7393,5797,2841,851,289,43,17,1,1
%N A110180 Triangle of generalized central trinomial coefficients.
%C A110180 Rows sums are A110181. Diagonal sums are A110182. Columns include central trinomial coefficients A002426, A084601, A084603, A084605, A098264. T(n,k) = central coefficient (1 + x + kx^2)^n.
%H A110180 G. C. Greubel, <a href="/A110180/b110180.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A110180 Number triangle T(n, k) = Sum_{j=0..floor((n-k)/2)} C(n-k, j)*C(n-k-j, j)*k^j.
%e A110180 Rows begin
%e A110180   1;
%e A110180   1,  1;
%e A110180   1,  1,  1;
%e A110180   1,  3,  1,  1;
%e A110180   1,  7,  5,  1,  1;
%e A110180   1, 19, 13,  7,  1,  1;
%t A110180 T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n - k, j]*Binomial[n - k - j, j]*k^j, {j, 0, Floor[(n - k)/2]}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Mar 05 2017 *)
%K A110180 easy,nonn,tabl
%O A110180 0,8
%A A110180 _Paul Barry_, Jul 14 2005