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A384987 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 A384984.

%I A384987 #9 Jun 14 2025 10:36:06
%S A384987 1,1,0,1,1,0,1,2,9,0,1,3,20,133,0,1,4,33,320,3185,0,1,5,48,567,7920,
%T A384987 88521,0,1,6,65,880,14529,232832,2709625,0,1,7,84,1265,23360,448203,
%U A384987 7695232,59590189,0,1,8,105,1728,34785,752064,15740001,220228416,-2800437663,0
%N A384987 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 A384984.
%F A384987 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-6*n+5*j+k)^(j-1) * binomial(n,j) * b(n-j,3*j). Then A(n,k) = b(n,-k).
%e A384987 Square array begins:
%e A384987   1,     1,      1,      1,      1,       1, ...
%e A384987   0,     1,      2,      3,      4,       5, ...
%e A384987   0,     9,     20,     33,     48,      65, ...
%e A384987   0,   133,    320,    567,    880,    1265, ...
%e A384987   0,  3185,   7920,  14529,  23360,   34785, ...
%e A384987   0, 88521, 232832, 448203, 752064, 1164125, ...
%o A384987 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-6*n+5*j+k)^(j-1)*binomial(n, j)*b(n-j, 3*j)));
%o A384987 a(n, k) = b(n, -k);
%Y A384987 Columns k=0..1 give A000007, A384984.
%K A384987 sign,tabl
%O A384987 0,8
%A A384987 _Seiichi Manyama_, Jun 14 2025