cp's OEIS Frontend

A384865 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 A213092.

%I A384865 #15 Jun 12 2025 08:29:25
%S A384865 1,1,0,1,1,0,1,2,1,0,1,3,3,3,0,1,4,6,8,8,0,1,5,10,16,23,31,0,1,6,15,
%T A384865 28,48,84,120,0,1,7,21,45,87,171,327,511,0,1,8,28,68,145,308,664,1372,
%U A384865 2234,0,1,9,36,98,228,516,1192,2760,5980,9988,0
%N A384865 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 A213092.
%C A384865 A(54,1) = -4736158402689145255029229896601957.
%F A384865 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} binomial(-3*n+4*j+k-1,j-1) * b(n-j,j)/j. Then A(n,k) = b(n,-k).
%e A384865 Square array begins:
%e A384865   1,   1,   1,   1,    1,    1,    1, ...
%e A384865   0,   1,   2,   3,    4,    5,    6, ...
%e A384865   0,   1,   3,   6,   10,   15,   21, ...
%e A384865   0,   3,   8,  16,   28,   45,   68, ...
%e A384865   0,   8,  23,  48,   87,  145,  228, ...
%e A384865   0,  31,  84, 171,  308,  516,  822, ...
%e A384865   0, 120, 327, 664, 1192, 1995, 3186, ...
%o A384865 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-3*n+4*j+k-1, j-1)*b(n-j, j)/j));
%o A384865 a(n, k) = b(n, -k);
%Y A384865 Columns k=0..1 give A000007, A213092.
%K A384865 tabl,sign
%O A384865 0,8
%A A384865 _Seiichi Manyama_, Jun 11 2025