cp's OEIS Frontend

A384722 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 A384720.

%I A384722 #12 Jun 08 2025 10:49:56
%S A384722 1,1,0,1,1,0,1,2,7,0,1,3,16,118,0,1,4,27,278,3385,0,1,5,40,486,8008,
%T A384722 141556,0,1,6,55,748,14121,333482,7918489,0,1,7,72,1070,22000,587268,
%U A384722 18524980,561302470,0,1,8,91,1458,31945,916084,32452353,1303041350,48589734337,0
%N A384722 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 A384720.
%F A384722 A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} (n-j+k)^(j-1) * binomial(n,j) * A(n-j,3*j).
%e A384722 Square array begins:
%e A384722   1,      1,      1,      1,      1,       1, ...
%e A384722   0,      1,      2,      3,      4,       5, ...
%e A384722   0,      7,     16,     27,     40,      55, ...
%e A384722   0,    118,    278,    486,    748,    1070, ...
%e A384722   0,   3385,   8008,  14121,  22000,   31945, ...
%e A384722   0, 141556, 333482, 587268, 916084, 1334900, ...
%o A384722 (PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, (n-j+k)^(j-1)*binomial(n, j)*a(n-j, 3*j)));
%Y A384722 Columns k=0..1 give A000007, A384720.
%Y A384722 Cf. A380178, A384721.
%K A384722 nonn,tabl
%O A384722 0,8
%A A384722 _Seiichi Manyama_, Jun 08 2025