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A385062 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 A385059.

%I A385062 #8 Jun 16 2025 08:42:04
%S A385062 1,1,0,1,1,0,1,2,7,0,1,3,16,40,0,1,4,27,122,397,0,1,5,40,252,1408,
%T A385062 2336,0,1,6,55,436,3285,14242,-46601,0,1,7,72,680,6304,42528,50200,
%U A385062 -2661896,0,1,8,91,990,10765,95684,457803,-4177822,-125428775,0,1,9,112,1372,16992,184000,1415728,-595872,-290217568,-3724273664,0
%N A385062 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 A385059.
%F A385062 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,2*j). Then A(n,k) = b(n,-k).
%e A385062 Square array begins:
%e A385062   1,    1,     1,     1,     1,      1, ...
%e A385062   0,    1,     2,     3,     4,      5, ...
%e A385062   0,    7,    16,    27,    40,     55, ...
%e A385062   0,   40,   122,   252,   436,    680, ...
%e A385062   0,  397,  1408,  3285,  6304,  10765, ...
%e A385062   0, 2336, 14242, 42528, 95684, 184000, ...
%o A385062 (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, 2*j)));
%o A385062 a(n, k) = b(n, -k);
%Y A385062 Columns k=0..1 give A000007, A385059.
%K A385062 sign,tabl
%O A385062 0,8
%A A385062 _Seiichi Manyama_, Jun 16 2025