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A384900 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 A384895.

%I A384900 #13 Jun 12 2025 10:22:57
%S A384900 1,1,0,1,1,0,1,2,2,0,1,3,5,3,0,1,4,9,10,4,0,1,5,14,22,18,-3,0,1,6,20,
%T A384900 40,48,14,-50,0,1,7,27,65,101,72,-81,-237,0,1,8,35,98,185,200,-37,
%U A384900 -562,-872,0,1,9,44,140,309,436,174,-873,-2420,-2375,0
%N A384900 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 A384895.
%F A384900 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-2*n+3*j+k-1,j-1) * b(n-j,2*j)/j. Then A(n,k) = b(n,-k).
%e A384900 Square array begins:
%e A384900   1,  1,  1,  1,   1,   1, ...
%e A384900   0,  1,  2,  3,   4,   5, ...
%e A384900   0,  2,  5,  9,  14,  20, ...
%e A384900   0,  3, 10, 22,  40,  65, ...
%e A384900   0,  4, 18, 48, 101, 185, ...
%e A384900   0, -3, 14, 72, 200, 436, ...
%o A384900 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-2*n+3*j+k-1,j-1)*b(n-j,2*j)/j));
%o A384900 a(n, k) = b(n, -k);
%Y A384900 Columns k=0..1 give A000007, A384895.
%K A384900 sign,tabl
%O A384900 0,8
%A A384900 _Seiichi Manyama_, Jun 12 2025