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A384690 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 A384689.

%I A384690 #12 Jun 07 2025 08:21:24
%S A384690 1,1,0,1,1,0,1,2,7,0,1,3,16,106,0,1,4,27,254,2593,0,1,5,40,450,6328,
%T A384690 89796,0,1,6,55,700,11457,220362,4085029,0,1,7,72,1010,18256,402468,
%U A384690 10016860,232694806,0,1,8,91,1386,27025,648564,18326853,568220102,16053415249,0
%N A384690 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 A384689.
%F A384690 A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} (n+j+k)^(j-1) * binomial(n,j) * A(n-j,j).
%e A384690 Square array begins:
%e A384690   1,     1,      1,      1,      1,      1, ...
%e A384690   0,     1,      2,      3,      4,      5, ...
%e A384690   0,     7,     16,     27,     40,     55, ...
%e A384690   0,   106,    254,    450,    700,   1010, ...
%e A384690   0,  2593,   6328,  11457,  18256,  27025, ...
%e A384690   0, 89796, 220362, 402468, 648564, 972900, ...
%o A384690 (PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, (n+j+k)^(j-1)*binomial(n, j)*a(n-j, j)));
%Y A384690 Columns k=0..1 give A000007, A384689.
%Y A384690 Cf. A379168, A380178.
%K A384690 nonn,tabl
%O A384690 0,8
%A A384690 _Seiichi Manyama_, Jun 07 2025