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A384811 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 A384809.

%I A384811 #15 Jul 27 2025 17:59:59
%S A384811 1,1,0,1,1,0,1,2,5,0,1,3,12,25,0,1,4,21,80,153,0,1,5,32,171,656,-799,
%T A384811 0,1,6,45,304,1689,2432,-82787,0,1,7,60,485,3456,13443,-139712,
%U A384811 -2990343,0,1,8,77,720,6185,37184,-100755,-7039744,-98020367,0,1,9,96,1015,10128,79925,143104,-11110677,-267665152,-2473062911,0
%N A384811 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 A384809.
%F A384811 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-2*n+2*j+k)^(j-1) * binomial(n,j) * b(n-j,2*j). Then A(n,k) = b(n,-k).
%e A384811 Square array begins:
%e A384811   1,    1,    1,     1,     1,     1, ...
%e A384811   0,    1,    2,     3,     4,     5, ...
%e A384811   0,    5,   12,    21,    32,    45, ...
%e A384811   0,   25,   80,   171,   304,   485, ...
%e A384811   0,  153,  656,  1689,  3456,  6185, ...
%e A384811   0, -799, 2432, 13443, 37184, 79925, ...
%o A384811 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-2*n+2*j+k)^(j-1)*binomial(n, j)*b(n-j, 2*j)));
%o A384811 a(n, k) = b(n, -k);
%Y A384811 Columns k=0..1 give A000007, A384809.
%K A384811 sign,tabl
%O A384811 0,8
%A A384811 _Seiichi Manyama_, Jun 10 2025