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A384751 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 A384749.

%I A384751 #10 Jun 09 2025 10:33:51
%S A384751 1,1,0,1,1,0,1,2,5,0,1,3,12,73,0,1,4,21,176,1881,0,1,5,32,315,4496,
%T A384751 73281,0,1,6,45,496,8025,172672,3919453,0,1,7,60,725,12672,304803,
%U A384751 9107008,271474953,0,1,8,77,1008,18665,477504,15874605,622823168,23404227185,0
%N A384751 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 A384749.
%F A384751 A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} (2*n-2*j+k)^(j-1) * binomial(n,j) * A(n-j,2*j).
%e A384751 Square array begins:
%e A384751   1,     1,      1,      1,      1,      1, ...
%e A384751   0,     1,      2,      3,      4,      5, ...
%e A384751   0,     5,     12,     21,     32,     45, ...
%e A384751   0,    73,    176,    315,    496,    725, ...
%e A384751   0,  1881,   4496,   8025,  12672,  18665, ...
%e A384751   0, 73281, 172672, 304803, 477504, 699925, ...
%o A384751 (PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, (2*n-2*j+k)^(j-1)*binomial(n, j)*a(n-j, 2*j)));
%Y A384751 Columns k=0..1 give A000007, A384749.
%K A384751 nonn,tabl
%O A384751 0,8
%A A384751 _Seiichi Manyama_, Jun 09 2025