cp's OEIS Frontend

A384802 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A213109.

%I A384802 #14 Jul 27 2025 17:59:46
%S A384802 1,1,0,1,1,0,1,2,3,0,1,3,8,22,0,1,4,15,62,233,0,1,5,24,126,696,3716,0,
%T A384802 1,6,35,220,1497,11082,77257,0,1,7,48,350,2768,24228,229756,2026606,0,
%U A384802 1,8,63,522,4665,46004,504657,5961846,63726497,0,1,9,80,742,7368,80100,969400,13042326,185814320,2333516392,0
%N A384802 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A213109.
%F A384802 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-3*n+3*j+k)^(j-1) * binomial(n,j) * b(n-j,j). Then A(n,k) = b(n,-k).
%e A384802 Square array begins:
%e A384802   1,    1,     1,     1,     1,     1, ...
%e A384802   0,    1,     2,     3,     4,     5, ...
%e A384802   0,    3,     8,    15,    24,    35, ...
%e A384802   0,   22,    62,   126,   220,   350, ...
%e A384802   0,  233,   696,  1497,  2768,  4665, ...
%e A384802   0, 3716, 11082, 24228, 46004, 80100, ...
%o A384802 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-3*n+3*j+k)^(j-1)*binomial(n, j)*b(n-j, j)));
%o A384802 a(n, k) = b(n, -k);
%Y A384802 Columns k=0..1 give A000007, A213109.
%K A384802 nonn,tabl
%O A384802 0,8
%A A384802 _Seiichi Manyama_, Jun 10 2025