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A384808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A384617.

%I A384808 #14 Jul 27 2025 17:59:56
%S A384808 1,1,0,1,1,0,1,2,5,0,1,3,12,13,0,1,4,21,56,-63,0,1,5,32,135,128,-2279,
%T A384808 0,1,6,45,256,753,-3888,-51167,0,1,7,60,425,2016,-1797,-135752,
%U A384808 -423387,0,1,8,77,648,4145,8224,-224775,-2099032,13717889,0,1,9,96,931,7392,31725,-256016,-5236809,3294432,885044593,0
%N A384808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A384617.
%F A384808 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-n+j+k)^(j-1) * binomial(n,j) * b(n-j,2*j). Then A(n,k) = b(n,-k).
%e A384808 Square array begins:
%e A384808   1,     1,     1,     1,    1,     1, ...
%e A384808   0,     1,     2,     3,    4,     5, ...
%e A384808   0,     5,    12,    21,   32,    45, ...
%e A384808   0,    13,    56,   135,  256,   425, ...
%e A384808   0,   -63,   128,   753, 2016,  4145, ...
%e A384808   0, -2279, -3888, -1797, 8224, 31725, ...
%o A384808 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-n+j+k)^(j-1)*binomial(n, j)*b(n-j, 2*j)));
%o A384808 a(n, k) = b(n, -k);
%Y A384808 Columns k=0..1 give A000007, A384617.
%K A384808 sign,tabl
%O A384808 0,8
%A A384808 _Seiichi Manyama_, Jun 10 2025