cp's OEIS Frontend

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

%I A084486 #19 Jan 24 2018 01:21:21
%S A084486 1,32,522,5776,50600,380424,2570932,16073600,94748400,533515240,
%T A084486 2896652396,15268777440,78544641448,395875164104,1960998472260,
%U A084486 9570684204544,46112171619296,219682468794600,1036237335593500
%N A084486 Number of 4 X n 0-1 matrices which have n+3 1's and have no zero rows or zero columns.
%C A084486 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.
%H A084486 M. Janjic and B. Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013
%H A084486 M. Janjic, B. Petkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic/janjic45.html">A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers</a>, J. Int. Seq. 17 (2014) # 14.3.5
%F A084486 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
%F A084486 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
%p A084486 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);
%t A084486 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));
%t A084486 Array[a, 20] (* _Jean-François Alcover_, Nov 10 2017, after _Vladeta Jovovic_ *)
%Y A084486 Cf. A001787.
%Y A084486 Cf. A084485, A055602, A055603.
%K A084486 nonn,easy
%O A084486 1,2
%A A084486 _W. Edwin Clark_, May 27 2003
%E A084486 Comment corrected by _W. Edwin Clark_, Sep 24 2012