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A384813 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 A384810.

%I A384813 #15 Jul 27 2025 18:00:07
%S A384813 1,1,0,1,1,0,1,2,5,0,1,3,12,37,0,1,4,21,104,417,0,1,5,32,207,1280,
%T A384813 4761,0,1,6,45,352,2769,17392,33313,0,1,7,60,545,5088,42363,213688,
%U A384813 -1509339,0,1,8,77,792,8465,85344,656505,-472456,-135791359,0,1,9,96,1099,13152,153325,1521904,6181815,-254502688,-8149132943,0
%N A384813 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 A384810.
%F A384813 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-3*n+3*j+k)^(j-1) * binomial(n,j) * b(n-j,2*j). Then A(n,k) = b(n,-k).
%e A384813 Square array begins:
%e A384813   1,    1,     1,     1,     1,      1, ...
%e A384813   0,    1,     2,     3,     4,      5, ...
%e A384813   0,    5,    12,    21,    32,     45, ...
%e A384813   0,   37,   104,   207,   352,    545, ...
%e A384813   0,  417,  1280,  2769,  5088,   8465, ...
%e A384813   0, 4761, 17392, 42363, 85344, 153325, ...
%o A384813 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-3*n+3*j+k)^(j-1)*binomial(n, j)*b(n-j, 2*j)));
%o A384813 a(n, k) = b(n, -k);
%Y A384813 Columns k=0..1 give A000007, A384810.
%Y A384813 Cf. A384808, A384811.
%K A384813 sign,tabl
%O A384813 0,8
%A A384813 _Seiichi Manyama_, Jun 10 2025