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A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2*k*x + ((k-2)*x)^2 + (1 - k*x) * sqrt(1 - 2*k*x + ((k-2)*x)^2)).

%I A331791 #55 Aug 27 2025 09:13:44
%S A331791 1,1,0,1,2,-3,1,4,3,0,1,6,15,4,10,1,8,33,56,5,0,1,10,57,180,210,6,-35,
%T A331791 1,12,87,400,985,792,7,0,1,14,123,740,2810,5418,3003,8,126,1,16,165,
%U A331791 1224,6285,19824,29953,11440,9,0,1,18,213,1876,12130,53550,140497,166344,43758,10,-462
%N A331791 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 2/(1 - 2*k*x + ((k-2)*x)^2 + (1 - k*x) * sqrt(1 - 2*k*x + ((k-2)*x)^2)).
%H A331791 Seiichi Manyama, <a href="/A331791/b331791.txt">Antidiagonals n = 0..139, flattened</a>
%F A331791 T(n,k) = Sum_{j=0..n} (k-1)^j * binomial(n+1,j) * binomial(n+1,j+1).
%F A331791 n * (n+2) * T(n,k) = (n+1) * (k * (2*n+1) * T(n-1,k) - (k-2)^2 * n * T(n-2,k)) for n > 1.
%F A331791 T(n,k) = Sum_{j=0..floor(n/2)} (k-1)^j * k^(n-2*j) * binomial(n+1,n-2*j) * binomial(2*j+1,j). - _Seiichi Manyama_, Aug 24 2025
%F A331791 From _Seiichi Manyama_, Aug 27 2025: (Start)
%F A331791 T(n,k) = [x^n] (1+k*x+(k-1)*x^2)^(n+1).
%F A331791 For k != 1, e.g.f. of column k: exp(k*x) * BesselI(1, 2*sqrt(k-1)*x) / sqrt(k-1), with offset 1. (End)
%e A331791 Square array begins:
%e A331791    1, 1,   1,    1,     1,     1, ...
%e A331791    0, 2,   4,    6,     8,    10, ...
%e A331791   -3, 3,  15,   33,    57,    87, ...
%e A331791    0, 4,  56,  180,   400,   740, ...
%e A331791   10, 5, 210,  985,  2810,  6285, ...
%e A331791    0, 6, 792, 5418, 19824, 53550, ...
%t A331791 T[n_, k_] := Sum[If[k==1 && j==0, 1, (k-1)^j] * Binomial[n + 1, j] * Binomial[n + 1, j + 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 05 2021 *)
%Y A331791 Columns k=1..5 give A000027(n+1), A001791(n+1), A050151(n+1), A331792, A331793.
%Y A331791 T(n,n+1) gives A331794.
%Y A331791 Cf. A307883, A331511, A331514, A331795.
%K A331791 sign,tabl,changed
%O A331791 0,5
%A A331791 _Seiichi Manyama_, Jan 26 2020