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A385020 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 A385016.

%I A385020 #10 Jun 15 2025 09:58:16
%S A385020 1,1,0,1,1,0,1,2,5,0,1,3,11,3,0,1,4,18,16,-51,0,1,5,26,40,-71,-190,0,
%T A385020 1,6,35,76,-45,-452,-401,0,1,7,45,125,43,-702,-1683,3672,0,1,8,56,188,
%U A385020 210,-836,-3784,4336,51925,0,1,9,68,266,474,-729,-6480,144,108645,151539,0
%N A385020 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 A385016.
%F A385020 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-n+j+k-1,j-1) * b(n-j,4*j)/j. Then A(n,k) = b(n,-k).
%e A385020 Square array begins:
%e A385020   1,    1,     1,     1,     1,     1,      1, ...
%e A385020   0,    1,     2,     3,     4,     5,      6, ...
%e A385020   0,    5,    11,    18,    26,    35,     45, ...
%e A385020   0,    3,    16,    40,    76,   125,    188, ...
%e A385020   0,  -51,   -71,   -45,    43,   210,    474, ...
%e A385020   0, -190,  -452,  -702,  -836,  -729,   -234, ...
%e A385020   0, -401, -1683, -3784, -6480, -9360, -11800, ...
%o A385020 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+j+k-1, j-1)*b(n-j, 4*j)/j));
%o A385020 a(n, k) = b(n, -k);
%Y A385020 Columns k=0..1 give A000007, A385016.
%K A385020 sign,tabl
%O A385020 0,8
%A A385020 _Seiichi Manyama_, Jun 15 2025