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A384859 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 A384855.

%I A384859 #15 Jun 12 2025 08:28:42
%S A384859 1,1,0,1,1,0,1,2,7,0,1,3,16,10,0,1,4,27,62,-503,0,1,5,40,162,-632,
%T A384859 -8564,0,1,6,55,316,-135,-20758,-103751,0,1,7,72,530,1264,-31572,
%U A384859 -413900,3479554,0,1,8,91,810,3865,-34316,-919647,2636678,327940225,0,1,9,112,1162,7992,-20500,-1552472,-5475222,679001872,8613464536,0
%N A384859 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 A384855.
%F A384859 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-n+j+k)^(j-1) * binomial(n,j) * b(n-j,3*j). Then A(n,k) = b(n,-k).
%e A384859 Square array begins:
%e A384859   1,     1,      1,      1,      1,      1, ...
%e A384859   0,     1,      2,      3,      4,      5, ...
%e A384859   0,     7,     16,     27,     40,     55, ...
%e A384859   0,    10,     62,    162,    316,    530, ...
%e A384859   0,  -503,   -632,   -135,   1264,   3865, ...
%e A384859   0, -8564, -20758, -31572, -34316, -20500, ...
%o A384859 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-n+j+k)^(j-1)*binomial(n, j)*b(n-j, 3*j)));
%o A384859 a(n, k) = b(n, -k);
%Y A384859 Columns k=0..1 give A000007, A384855.
%Y A384859 Cf. A384801, A384808.
%K A384859 sign,tabl
%O A384859 0,8
%A A384859 _Seiichi Manyama_, Jun 10 2025