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A384742 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 A384740.

%I A384742 #11 Jun 09 2025 10:34:02
%S A384742 1,1,0,1,1,0,1,2,3,0,1,3,8,34,0,1,4,15,86,665,0,1,5,24,162,1656,20556,
%T A384742 0,1,6,35,268,3081,49802,901417,0,1,7,48,410,5072,90588,2132476,
%U A384742 52455250,0,1,8,63,594,7785,146484,3792177,121703094,3885229665,0
%N A384742 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 A384740.
%F A384742 A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} (3*n-3*j+k)^(j-1) * binomial(n,j) * A(n-j,j).
%e A384742 Square array begins:
%e A384742   1,     1,     1,     1,      1,      1, ...
%e A384742   0,     1,     2,     3,      4,      5, ...
%e A384742   0,     3,     8,    15,     24,     35, ...
%e A384742   0,    34,    86,   162,    268,    410, ...
%e A384742   0,   665,  1656,  3081,   5072,   7785, ...
%e A384742   0, 20556, 49802, 90588, 146484, 221900, ...
%o A384742 (PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, (3*n-3*j+k)^(j-1)*binomial(n, j)*a(n-j, j)));
%Y A384742 Columns k=0..1 give A000007, A384740.
%Y A384742 Cf. A380178, A384741.
%K A384742 nonn,tabl
%O A384742 0,8
%A A384742 _Seiichi Manyama_, Jun 08 2025