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A384804 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 A384803.

%I A384804 #14 Jul 27 2025 17:59:53
%S A384804 1,1,0,1,1,0,1,2,3,0,1,3,8,28,0,1,4,15,74,365,0,1,5,24,144,1008,7456,
%T A384804 0,1,6,35,244,2037,20242,198967,0,1,7,48,380,3584,40848,535936,
%U A384804 6600448,0,1,8,63,558,5805,72484,1076427,17641290,260641817,0,1,9,80,784,8880,119200,1909120,35239872,693025024,11805179392,0
%N A384804 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 A384803.
%F A384804 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-4*n+4*j+k)^(j-1) * binomial(n,j) * b(n-j,j). Then A(n,k) = b(n,-k).
%e A384804 Square array begins:
%e A384804   1,    1,     1,     1,     1,      1, ...
%e A384804   0,    1,     2,     3,     4,      5, ...
%e A384804   0,    3,     8,    15,    24,     35, ...
%e A384804   0,   28,    74,   144,   244,    380, ...
%e A384804   0,  365,  1008,  2037,  3584,   5805, ...
%e A384804   0, 7456, 20242, 40848, 72484, 119200, ...
%o A384804 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-4*n+4*j+k)^(j-1)*binomial(n, j)*b(n-j, j)));
%o A384804 a(n, k) = b(n, -k);
%Y A384804 Columns k=0..1 give A000007, A384803.
%K A384804 nonn,tabl
%O A384804 0,8
%A A384804 _Seiichi Manyama_, Jun 10 2025