cp's OEIS Frontend

A381512 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = (2*n+k)!/k! * [x^(2*n+k)] sinh(x)^k.

%I A381512 #41 May 12 2025 11:28:37
%S A381512 1,1,0,1,1,0,1,4,1,0,1,10,16,1,0,1,20,91,64,1,0,1,35,336,820,256,1,0,
%T A381512 1,56,966,5440,7381,1024,1,0,1,84,2352,24970,87296,66430,4096,1,0,1,
%U A381512 120,5082,90112,631631,1397760,597871,16384,1,0,1,165,10032,273988,3331328,15857205,22368256,5380840,65536,1,0
%N A381512 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = (2*n+k)!/k! * [x^(2*n+k)] sinh(x)^k.
%F A381512 G.f. of column k: 1/Product_{j=0..floor(k/2)} (1 - (k-2*j)^2*x).
%F A381512 A(n,k) = k^2 * A(n-1,k) + A(n,k-2) for k > 1.
%F A381512 A(n,k) = (1/(2^k*k!)) * Sum_{j=0..k} (-1)^j * (k-2*j)^(2*n+k) * binomial(k,j).
%e A381512 Square array begins:
%e A381512   1, 1,    1,     1,       1,        1, ...
%e A381512   0, 1,    4,    10,      20,       35, ...
%e A381512   0, 1,   16,    91,     336,      966, ...
%e A381512   0, 1,   64,   820,    5440,    24970, ...
%e A381512   0, 1,  256,  7381,   87296,   631631, ...
%e A381512   0, 1, 1024, 66430, 1397760, 15857205, ...
%o A381512 (PARI) a(n, k) = (2*n+k)!/k!*polcoef(sinh(x+x*O(x^(2*n+k)))^k, 2*n+k);
%Y A381512 Columns k=0..7 give A000007, A000012, A000302, A002452(n+1), A166984, A002453, 4^n * A002451(n), A381513.
%Y A381512 Main diagonal gives A383837.
%Y A381512 Cf. A136630, A160562, A286899.
%K A381512 nonn,tabl
%O A381512 0,8
%A A381512 _Seiichi Manyama_, May 11 2025