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A230880 Number of 2-packed matrices with exactly n nonzero entries.

%I A230880 #16 Oct 08 2017 08:54:56
%S A230880 1,2,8,80,1120,20544,463744,12422656,384947200,13541822464,
%T A230880 533049493504,23210958688256,1107652218822656,57482801016422400,
%U A230880 3223015475535380480,194157345516262588416,12505948470244176953344,857670052436844788318208,62395270194815987194789888
%N A230880 Number of 2-packed matrices with exactly n nonzero entries.
%C A230880 A k-packed matrix of size n X n is a matrix with entries in the alphabet A_k = {0,1, ..., k} such that each row and each column contains at least one nonzero entry.
%H A230880 Andrew Howroyd, <a href="/A230880/b230880.txt">Table of n, a(n) for n = 0..100</a>
%H A230880 H. Cheballah, S. Giraudo, R. Maurice, <a href="http://arxiv.org/abs/1306.6605">Combinatorial Hopf algebra structure on packed square matrices</a>, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
%F A230880 Cheballah et al. give an explicit formula.
%F A230880 From _Andrew Howroyd_, Sep 20 2017: (Start)
%F A230880 a(n) = Sum_{r=1..n} Sum_{i=0..r} Sum_{j=0..r} (-1)^(i+j) * binomial(r,i) * binomial(r,j) * binomial(i*j,n) * 2^n.
%F A230880 a(n) = 2^n * A104602(n).
%F A230880 (End)
%t A230880 b[n_] := Sum[StirlingS1[n, k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}], {k, 0, n}]/n!;
%t A230880 a[n_] := 2^n*b[n];
%t A230880 Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Oct 08 2017, translated from PARI *)
%o A230880 (PARI) \\ here b(n) is A104602.
%o A230880 b(n) = {sum(m=0, n, sum(k=0, n, stirling(n,k,1) * m!^2 * stirling(k,m,2)^2)) / n!}
%o A230880 a(n) = 2^n * b(n); \\ _Andrew Howroyd_, Sep 20 2017
%Y A230880 Cf. A048291, A055599, A230878, A230879, A104602.
%K A230880 nonn
%O A230880 0,2
%A A230880 _N. J. A. Sloane_, Nov 09 2013
%E A230880 Terms a(9) and beyond from _Andrew Howroyd_, Sep 20 2017