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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).

%I A230878 #16 Oct 08 2017 08:56:43
%S A230878 1,0,2,0,0,8,32,16,0,0,0,48,720,2880,4992,4608,2304,512,0,0,0,0,384,
%T A230878 13824,143872,739328,2320896,4964352,7659520,8749056,7421952,4587520,
%U A230878 1966080,524288,65536,0,0,0,0,0,3840,268800,5504000,57068800,372416000
%N 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).
%C A230878 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 A230878 Andrew Howroyd, <a href="/A230878/b230878.txt">Table of n, a(n) for n = 0..662</a> (rows 0..12 flattened)
%H A230878 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 A230878 From _Andrew Howroyd_, Sep 20 2017: (Start)
%F A230878 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.
%F A230878 T(n, k) = 0 for n > k.
%F A230878 T(n, n) = A000165(n).
%F A230878 (End)
%e A230878 Triangle begins:
%e A230878 1
%e A230878 0 2
%e A230878 0 0 8 32 16
%e A230878 0 0 0 48 720 2880 4992 4608 2304 512
%e A230878 ...
%t A230878 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 A230878 T[n_, k_] := p[2, n, k];
%t A230878 Table[T[n, k], {n, 0, 5}, {k, 0, n^2}] // Flatten (* _Jean-François Alcover_, Oct 08 2017, translated from PARI *)
%o A230878 (PARI) \\ T(n,k) = p(2,n,k) (see Cheballah et al. ref).
%o A230878 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))}
%o A230878 for (n=0,5, for(k=0,n^2, print1(p(2,n,k), ", ")); print); \\ _Andrew Howroyd_, Sep 20 2017
%Y A230878 Row sums are A230879.
%Y A230878 Column sums are A230880.
%Y A230878 Cf. A000165, A055599.
%K A230878 nonn,tabf
%O A230878 0,3
%A A230878 _N. J. A. Sloane_, Nov 09 2013
%E A230878 Terms a(18) and beyond from _Andrew Howroyd_, Sep 20 2017