cp's OEIS Frontend

A384899 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 A384894.

%I A384899 #14 Jun 12 2025 10:23:04
%S A384899 1,1,0,1,1,0,1,2,2,0,1,3,5,1,0,1,4,9,6,-4,0,1,5,14,16,-2,-14,0,1,6,20,
%T A384899 32,12,-32,-30,0,1,7,27,55,45,-39,-103,12,0,1,8,35,86,105,-12,-211,
%U A384899 -100,330,0,1,9,44,126,201,81,-318,-411,552,1139,0,1,10,54,176,343,282,-350,-956,342,3038,2226,0
%N A384899 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 A384894.
%F A384899 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,2*j)/j. Then A(n,k) = b(n,-k).
%e A384899 Square array begins:
%e A384899   1,   1,   1,   1,   1,   1, ...
%e A384899   0,   1,   2,   3,   4,   5, ...
%e A384899   0,   2,   5,   9,  14,  20, ...
%e A384899   0,   1,   6,  16,  32,  55, ...
%e A384899   0,  -4,  -2,  12,  45, 105, ...
%e A384899   0, -14, -32, -39, -12,  81, ...
%o A384899 (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,2*j)/j));
%o A384899 a(n, k) = b(n, -k);
%Y A384899 Columns k=0..1 give A000007, A384894.
%K A384899 sign,tabl
%O A384899 0,8
%A A384899 _Seiichi Manyama_, Jun 12 2025