A230878 Irregular triangle read by rows: T(n,k) = number of 2-packed n X n matrices with exactly k nonzero entries (0 <= k <= n^2).

1, 0, 2, 0, 0, 8, 32, 16, 0, 0, 0, 48, 720, 2880, 4992, 4608, 2304, 512, 0, 0, 0, 0, 384, 13824, 143872, 739328, 2320896, 4964352, 7659520, 8749056, 7421952, 4587520, 1966080, 524288, 65536, 0, 0, 0, 0, 0, 3840, 268800, 5504000, 57068800, 372416000

Offset

0,3

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..662 (rows 0..12 flattened)

H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.

Formula

From Andrew Howroyd, Sep 20 2017: (Start)

T(n, k) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * binomial(n,i) * binomial(n,j) * binomial(i*j,k) * 2^k.

T(n, k) = 0 for n > k.

T(n, n) = A000165(n).

(End)

Example

Triangle begins:
1
0 2
0 0 8 32 16
0 0 0 48 720 2880 4992 4608 2304 512
...

Mathematica

p[k_, n_, l_] := Sum[(-1)^(i+j)*Binomial[n, i]*Binomial[n, j]*Binomial[i*j, l]*k^l, {i, 0, n}, {j, 0, n}];
T[n_, k_] := p[2, n, k];
Table[T[n, k], {n, 0, 5}, {k, 0, n^2}] // Flatten (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

Prog

(PARI) \\ T(n, k) = p(2, n, k) (see Cheballah et al. ref).
p(k, n, l) = {sum(i=0, n, sum(j=0, n, (-1)^(i+j) * binomial(n, i) * binomial(n, j) * binomial(i*j, l) * k^l))}
for (n=0, 5, for(k=0, n^2, print1(p(2, n, k), ", ")); print); \\ Andrew Howroyd, Sep 20 2017

Crossrefs

Row sums are A230879.

Column sums are A230880.

Cf. A000165, A055599.

Sequence in context: A134414 A113036 A000425 * A349369 A349352 A323896

Adjacent sequences: A230875 A230876 A230877 * A230879 A230880 A230881

Keyword

nonn,tabf

Author

N. J. A. Sloane, Nov 09 2013

Extensions

Terms a(18) and beyond from Andrew Howroyd, Sep 20 2017

Status

approved