A084486 Number of 4 X n 0-1 matrices which have n+3 1's and have no zero rows or zero columns.

1, 32, 522, 5776, 50600, 380424, 2570932, 16073600, 94748400, 533515240, 2896652396, 15268777440, 78544641448, 395875164104, 1960998472260, 9570684204544, 46112171619296, 219682468794600, 1036237335593500

Offset

1,2

Comments

This is the number of spanning subgraphs of the complete bipartite graph K(4,n) which have n+3 edges and no isolated vertices. If the subgraphs are also connected then they are spanning trees. The number of spanning trees in K(m,n) is known. See A001787.

Links

Table of n, a(n) for n=1..19.

M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013

M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5

Formula

n/48*((27*4^n-32*3^n+6*2^n)*n^2+(-9*4^n+32*3^n-18*2^n)*n+(-6*4^n+12*2^n)). - Vladeta Jovovic, May 28 2003

G.f.: x * (1 -4*x -40*x^2 +44*x^3 +2885*x^4 -19624*x^5 +59014*x^6 -97728*x^7 +98064*x^8 -67200*x^9 +28800*x^10) / ((3*x-1)^4*(2*x-1)^4*(4*x-1)^4). - Alois P. Heinz, Sep 24 2012

Maple

with(LinearAlgebra): num1s := (M, m, n)->add(ListTools[Flatten](convert(M, listlist))[j], j=1..m*n): binrows := n->[seq(convert(i+2^n, base, 2)[1..n], i=1..2^n-1)]; a := proc(n) local A, L, i, j, k, el, S, M: S := 0: L := binrows(n): for i from 1 to 2^n-1 do for j from 1 to 2^n-1 do for k from 1 to 2^n-1 do for el from 1 to 2^n-1 do A := Matrix([L[i], L[j], L[k], L[el]]); if num1s(A, 4, n)=n+3 and (not has(Matrix([1, 1, 1, 1]).A, 0)) then S := S+1; end if; od; od; od; od; S; end proc: seq (a(n), n=1..2);

Mathematica

a[n_] := n/48*((27*4^n - 32*3^n + 6*2^n)*n^2 + (-9*4^n + 32*3^n - 18*2^n)*n + (-6*4^n + 12*2^n));
Array[a, 20] (* Jean-François Alcover, Nov 10 2017, after Vladeta Jovovic *)

Crossrefs

Cf. A001787.

Cf. A084485, A055602, A055603.

Sequence in context: A366551 A146124 A125489 * A161640 A161987 A162379

Adjacent sequences: A084483 A084484 A084485 * A084487 A084488 A084489

Keyword

nonn,easy

Author

W. Edwin Clark, May 27 2003

Extensions

Comment corrected by W. Edwin Clark, Sep 24 2012

Status

approved