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A384862 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 A384858.

%I A384862 #14 Jun 12 2025 08:29:34
%S A384862 1,1,0,1,1,0,1,2,7,0,1,3,16,136,0,1,4,27,314,3781,0,1,5,40,540,8944,
%T A384862 163216,0,1,6,55,820,15741,383282,9103699,0,1,7,72,1160,24448,672768,
%U A384862 21329920,646696576,0,1,8,91,1566,35365,1045924,37392543,1504825562,55084545289,0
%N A384862 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 A384858.
%F A384862 Let b(n,k) = 0^n if n*k=0, otherwise b(n,k) = (-1)^n * k * Sum_{j=1..n} (-8*n+8*j+k)^(j-1) * binomial(n,j) * b(n-j,3*j). Then A(n,k) = b(n,-k).
%e A384862 Square array begins:
%e A384862   1,      1,      1,      1,       1,       1, ...
%e A384862   0,      1,      2,      3,       4,       5, ...
%e A384862   0,      7,     16,     27,      40,      55, ...
%e A384862   0,    136,    314,    540,     820,    1160, ...
%e A384862   0,   3781,   8944,  15741,   24448,   35365, ...
%e A384862   0, 163216, 383282, 672768, 1045924, 1518800, ...
%o A384862 (PARI) b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, (-8*n+8*j+k)^(j-1)*binomial(n, j)*b(n-j, 3*j)));
%o A384862 a(n, k) = b(n, -k);
%Y A384862 Columns k=0..1 give A000007, A384858.
%K A384862 nonn,tabl
%O A384862 0,8
%A A384862 _Seiichi Manyama_, Jun 10 2025