cp's OEIS Frontend

A384861 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 A384857.

%I A384861 #17 Jun 12 2025 08:29:55
%S A384861 1,1,0,1,1,0,1,2,7,0,1,3,16,46,0,1,4,27,134,361,0,1,5,40,270,1384,
%T A384861 -6284,0,1,6,55,460,3321,-2518,-632951,0,1,7,72,710,6448,18468,
%U A384861 -1223180,-31583474,0,1,8,91,1026,11065,65524,-1591407,-72713338,-1484748191,0
%N A384861 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 A384857.
%F A384861 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,3*j). Then A(n,k) = b(n,-k).
%e A384861 Square array begins:
%e A384861   1,     1,     1,     1,     1,      1, ...
%e A384861   0,     1,     2,     3,     4,      5, ...
%e A384861   0,     7,    16,    27,    40,     55, ...
%e A384861   0,    46,   134,   270,   460,    710, ...
%e A384861   0,   361,  1384,  3321,  6448,  11065, ...
%e A384861   0, -6284, -2518, 18468, 65524, 149300, ...
%o A384861 (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, 3*j)));
%o A384861 a(n, k) = b(n, -k);
%Y A384861 Columns k=0..1 give A000007, A384857.
%Y A384861 Cf. A384802, A384813.
%K A384861 sign,tabl
%O A384861 0,8
%A A384861 _Seiichi Manyama_, Jun 10 2025