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A384741 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 A384739.

%I A384741 #11 Jun 09 2025 10:34:06
%S A384741 1,1,0,1,1,0,1,2,3,0,1,3,8,28,0,1,4,15,74,461,0,1,5,24,144,1200,11776,
%T A384741 0,1,6,35,244,2325,29842,421207,0,1,7,48,380,3968,56688,1040896,
%U A384741 19832128,0,1,8,63,558,6285,95524,1933227,47948490,1179482201,0
%N A384741 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 A384739.
%F A384741 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,j).
%e A384741 Square array begins:
%e A384741   1,     1,     1,     1,     1,      1, ...
%e A384741   0,     1,     2,     3,     4,      5, ...
%e A384741   0,     3,     8,    15,    24,     35, ...
%e A384741   0,    28,    74,   144,   244,    380, ...
%e A384741   0,   461,  1200,  2325,  3968,   6285, ...
%e A384741   0, 11776, 29842, 56688, 95524, 150400, ...
%o A384741 (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, j)));
%Y A384741 Columns k=0..1 give A000007, A384739.
%Y A384741 Cf. A380178, A384742.
%K A384741 nonn,tabl
%O A384741 0,8
%A A384741 _Seiichi Manyama_, Jun 08 2025