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A384692 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 A384691.

%I A384692 #9 Jun 07 2025 08:21:30
%S A384692 1,1,0,1,1,0,1,2,7,0,1,3,16,112,0,1,4,27,266,2989,0,1,5,40,468,7168,
%T A384692 115136,0,1,6,55,724,12789,275842,5899159,0,1,7,72,1040,20128,493248,
%U A384692 14058520,381657928,0,1,8,91,1422,29485,780164,25060203,903187826,30082660633,0
%N A384692 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 A384691.
%F A384692 A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} (n+k)^(j-1) * binomial(n,j) * A(n-j,2*j).
%e A384692 Square array begins:
%e A384692   1,      1,      1,      1,      1,       1, ...
%e A384692   0,      1,      2,      3,      4,       5, ...
%e A384692   0,      7,     16,     27,     40,      55, ...
%e A384692   0,    112,    266,    468,    724,    1040, ...
%e A384692   0,   2989,   7168,  12789,  20128,   29485, ...
%e A384692   0, 115136, 275842, 493248, 780164, 1151200, ...
%o A384692 (PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, (n+k)^(j-1)*binomial(n, j)*a(n-j, 2*j)));
%Y A384692 Columns k=0..1 give A000007, A384691.
%K A384692 nonn,tabl
%O A384692 0,8
%A A384692 _Seiichi Manyama_, Jun 07 2025