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A385063 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 A385060.

%I A385063 #8 Jun 16 2025 08:41:38
%S A385063 1,1,0,1,1,0,1,2,9,0,1,3,20,43,0,1,4,33,140,125,0,1,5,48,297,1080,
%T A385063 -6279,0,1,6,65,520,3189,-3568,-412025,0,1,7,84,815,6800,18003,
%U A385063 -828668,-9060911,0,1,8,105,1188,12285,70464,-1033749,-25887384,-98234103,0
%N A385063 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 A385060.
%F A385063 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-n+k)^(j-1) * binomial(n,j) * b(n-j,3*j). Then A(n,k) = b(n,-k).
%e A385063 Square array begins:
%e A385063   1,     1,     1,     1,     1,      1, ...
%e A385063   0,     1,     2,     3,     4,      5, ...
%e A385063   0,     9,    20,    33,    48,     65, ...
%e A385063   0,    43,   140,   297,   520,    815, ...
%e A385063   0,   125,  1080,  3189,  6800,  12285, ...
%e A385063   0, -6279, -3568, 18003, 70464, 168125, ...
%o A385063 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-n+k)^(j-1)*binomial(n, j)*b(n-j, 3*j)));
%o A385063 a(n, k) = b(n, -k);
%Y A385063 Columns k=0..1 give A000007, A385060.
%K A385063 sign,tabl
%O A385063 0,8
%A A385063 _Seiichi Manyama_, Jun 16 2025