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A384758 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 A384757.

%I A384758 #11 Jun 09 2025 10:33:38
%S A384758 1,1,0,1,-1,0,1,-2,-1,0,1,-3,0,14,0,1,-4,3,34,9,0,1,-5,8,54,-88,-1516,
%T A384758 0,1,-6,15,68,-327,-3402,4345,0,1,-7,24,70,-720,-4908,30532,507870,0,
%U A384758 1,-8,35,54,-1255,-5044,84321,1027402,-4984063,0
%N A384758 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 A384757.
%F A384758 A(n,0) = 0^n; A(n,k) = (-1)^n * k * Sum_{j=0..n} (n-j+k)^(j-1) * binomial(n,j) * A(n-j,j).
%e A384758 Square array begins:
%e A384758   1,     1,     1,     1,     1,     1, ...
%e A384758   0,    -1,    -2,    -3,    -4,    -5, ...
%e A384758   0,    -1,     0,     3,     8,    15, ...
%e A384758   0,    14,    34,    54,    68,    70, ...
%e A384758   0,     9,   -88,  -327,  -720, -1255, ...
%e A384758   0, -1516, -3402, -4908, -5044, -2700, ...
%o A384758 (PARI) a(n, k) = if(k==0, 0^n, (-1)^n*k*sum(j=0, n, (n-j+k)^(j-1)*binomial(n, j)*a(n-j, j)));
%Y A384758 Columns k=0..1 give A000007, A384757.
%Y A384758 Cf. A384761.
%K A384758 sign,tabl
%O A384758 0,8
%A A384758 _Seiichi Manyama_, Jun 09 2025