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%I #17 Nov 15 2018 08:39:58
%S 1,1,3,15,107,991,11267,151721,2360375,41650861,821881709,17932031225,
%T 428630422697,11138928977049,312680873171465,9428701154866535,
%U 303957777464447449,10431949496859168189,379755239311735494421
%N Number of square matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.
%H G. C. Greubel, <a href="/A120732/b120732.txt">Table of n, a(n) for n = 0..250</a>
%F a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A048144(k).
%F G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1-x)^(-j)-1)^n.
%F a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.4670932578797312973586879293426... . - _Vaclav Kotesovec_, May 07 2014
%F In closed form, c = 2^(log(2)/2-2) / (log(2) * sqrt(Pi*(1-log(2)))). - _Vaclav Kotesovec_, May 03 2015
%F G.f.: Sum_{n>=0} (1-x)^n * (1 - (1-x)^n)^n. - _Paul D. Hanna_, Mar 26 2018
%e From _Gus Wiseman_, Nov 14 2018: (Start)
%e The a(3) = 15 matrices:
%e [3]
%e .
%e [2 0] [1 1] [1 1] [1 0] [1 0] [0 2] [0 1] [0 1]
%e [0 1] [1 0] [0 1] [1 1] [0 2] [1 0] [2 0] [1 1]
%e .
%e [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
%e [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
%e [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]
%e (End)
%t Table[1/n!*Sum[(-1)^(n-k)*StirlingS1[n,k]*Sum[(m!)^2*StirlingS2[k,m]^2,{m,0,k}],{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, May 07 2014 *)
%t multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n],2],n],Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]],{n,5}] (* _Gus Wiseman_, Nov 14 2018 *)
%Y Cf. A007716, A048291, A054976, A057149, A057150, A057151, A104601, A104602, A120733, A138178, A316983, A319616.
%K nonn
%O 0,3
%A _Vladeta Jovovic_, Aug 18 2006
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