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A321660 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose entries are all distinct.

%I A321660 #12 Nov 17 2018 21:11:05
%S A321660 1,1,1,5,5,9,45,49,85,125,233,273,417,529,745,2573,2861,4761,6837,
%T A321660 10489,14317,22637,28289,40041,52041,70177,88561,117605,234773,274761,
%U A321660 407469,553681,792613,1052525,1493033,1959009,3135537,3904129,5475673,7173725,9853325
%N A321660 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose entries are all distinct.
%H A321660 Andrew Howroyd, <a href="/A321660/b321660.txt">Table of n, a(n) for n = 0..1000</a>
%F A321660 a(n) = Sum_{k>=1} (k!*A000005(k) + (k+1)!*(A000005(k+1) - 2))*A008289(n,k) for n > 0. - _Andrew Howroyd_, Nov 17 2018
%e A321660 The a(5) = 9 matrices:
%e A321660   [5] [4 1] [3 2] [2 3] [1 4]
%e A321660 .
%e A321660   [4] [3] [2] [1]
%e A321660   [1] [2] [3] [4]
%t A321660 prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
%t A321660 multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
%t A321660 Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@Join@@prs2mat[#]]&]],{n,5}]
%o A321660 (PARI) seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, (n-1)!*numdiv(n-1) + n!*(numdiv(n) - 2)))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ _Andrew Howroyd_, Nov 16 2018
%Y A321660 Cf. A000005, A000219, A007716, A008289, A114736, A117433, A120733, A321645, A321653, A321655, A321659, A321661, A321662.
%K A321660 nonn
%O A321660 0,4
%A A321660 _Gus Wiseman_, Nov 15 2018
%E A321660 Terms a(11) and beyond from _Andrew Howroyd_, Nov 16 2018