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A230880
Number of 2-packed matrices with exactly n nonzero entries.
3
1, 2, 8, 80, 1120, 20544, 463744, 12422656, 384947200, 13541822464, 533049493504, 23210958688256, 1107652218822656, 57482801016422400, 3223015475535380480, 194157345516262588416, 12505948470244176953344, 857670052436844788318208, 62395270194815987194789888
OFFSET
0,2
COMMENTS
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.
LINKS
H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
FORMULA
Cheballah et al. give an explicit formula.
From Andrew Howroyd, Sep 20 2017: (Start)
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.
a(n) = 2^n * A104602(n).
(End)
MATHEMATICA
b[n_] := Sum[StirlingS1[n, k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}], {k, 0, n}]/n!;
a[n_] := 2^n*b[n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)
PROG
(PARI) \\ here b(n) is A104602.
b(n) = {sum(m=0, n, sum(k=0, n, stirling(n, k, 1) * m!^2 * stirling(k, m, 2)^2)) / n!}
a(n) = 2^n * b(n); \\ Andrew Howroyd, Sep 20 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 09 2013
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Sep 20 2017
STATUS
approved