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
Andrew Howroyd, Table of n, a(n) for n = 0..30
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.
a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * binomial(n,i) * binomial(n,j) * 3^(i*j). - Andrew Howroyd, Sep 20 2017
MATHEMATICA
p[k_, n_] := Sum[(-1)^(i + j)*Binomial[n, i]*Binomial[n, j]*(k + 1)^(i*j), {i, 0, n}, {j, 0, n}];
a[n_] := p[2, n];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)
PROG
(PARI) \\ here p(k, n) is number of k-packed matrices of size n.
p(k, n)={sum(i=0, n, sum(j=0, n, (-1)^(i+j) * binomial(n, i) * binomial(n, j) * (k+1)^(i*j)))}
a(n) = p(2, n); \\ Andrew Howroyd, Sep 20 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 09 2013
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, Sep 20 2017
STATUS
approved