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A385017 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 A385013.

%I A385017 #10 Jun 15 2025 09:57:04
%S A385017 1,1,0,1,1,0,1,2,2,0,1,3,5,4,0,1,4,9,12,9,0,1,5,14,25,30,19,0,1,6,20,
%T A385017 44,69,72,37,0,1,7,27,70,133,183,164,52,0,1,8,35,104,230,384,464,326,
%U A385017 -25,0,1,9,44,147,369,716,1060,1083,435,-630,0,1,10,54,200,560,1230,2125,2748,2139,-464,-3616,0
%N A385017 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 A385013.
%F A385017 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-2*n+2*j+k-1,j-1) * b(n-j,j)/j. Then A(n,k) = b(n,-k).
%e A385017 Square array begins:
%e A385017   1,  1,   1,   1,    1,    1,    1, ...
%e A385017   0,  1,   2,   3,    4,    5,    6, ...
%e A385017   0,  2,   5,   9,   14,   20,   27, ...
%e A385017   0,  4,  12,  25,   44,   70,  104, ...
%e A385017   0,  9,  30,  69,  133,  230,  369, ...
%e A385017   0, 19,  72, 183,  384,  716, 1230, ...
%e A385017   0, 37, 164, 464, 1060, 2125, 3893, ...
%o A385017 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-2*n+2*j+k-1, j-1)*b(n-j, j)/j));
%o A385017 a(n, k) = b(n, -k);
%Y A385017 Columns k=0..1 give A000007, A385013.
%Y A385017 Cf. A384976.
%K A385017 sign,tabl
%O A385017 0,8
%A A385017 _Seiichi Manyama_, Jun 15 2025