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A384901 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384896.

%I A384901 #12 Jun 12 2025 10:23:07
%S A384901 1,1,0,1,1,0,1,2,3,0,1,3,7,0,0,1,4,12,6,-23,0,1,5,18,19,-37,-51,0,1,6,
%T A384901 25,40,-33,-148,27,0,1,7,33,70,-1,-264,-186,920,0,1,8,42,110,70,-360,
%U A384901 -681,1588,5469,0,1,9,52,161,192,-384,-1446,1437,13469,4836,0
%N A384901 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384896.
%F A384901 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-n+2*j+k-1,j-1) * b(n-j,3*j)/j. Then A(n,k) = b(n,-k).
%e A384901 Square array begins:
%e A384901   1,   1,    1,    1,    1,    1, ...
%e A384901   0,   1,    2,    3,    4,    5, ...
%e A384901   0,   3,    7,   12,   18,   25, ...
%e A384901   0,   0,    6,   19,   40,   70, ...
%e A384901   0, -23,  -37,  -33,   -1,   70, ...
%e A384901   0, -51, -148, -264, -360, -384, ...
%o A384901 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+2*j+k-1,j-1)*b(n-j,3*j)/j));
%o A384901 a(n, k) = b(n, -k);
%Y A384901 Columns k=0..1 give A000007, A384896.
%K A384901 sign,tabl
%O A384901 0,8
%A A384901 _Seiichi Manyama_, Jun 12 2025