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A360660 Number of inequivalent n X n {0,1} matrices modulo permutation of the rows, with exactly n 1's.

%I A360660 #45 Apr 15 2025 14:11:28
%S A360660 1,1,4,20,133,1027,9259,94033,1062814,13176444,177427145,2573224238,
%T A360660 39924120823,658921572675,11513293227271,212109149134617,
%U A360660 4105637511110979,83238756058333277,1762856698153603049,38905470655863251479,892840913430059075405
%N A360660 Number of inequivalent n X n {0,1} matrices modulo permutation of the rows, with exactly n 1's.
%C A360660 Also the number of multisets of n words of length n over binary alphabet where the first letter occurs n times. E.g., a(2) = 4: {aa,bb}, {ab,ab}, {ab,ba}, {ba,ba}.
%H A360660 Alois P. Heinz, <a href="/A360660/b360660.txt">Table of n, a(n) for n = 0..200</a>
%F A360660 a(n) = A220886(n,n).
%F A360660 a(n) = [x^n] Product_{k=1..n} 1/(1 - x^k)^binomial(n,k). - _Vaclav Kotesovec_, Apr 15 2025
%e A360660 a(3) = 20: [111/000/000], [110/100/000], [110/010/000], [110/001/000], [101/100/000], [101/010/000], [101/001/000], [100/100/100], [100/100/010], [100/100/001], [100/011/000], [100/010/010], [100/010/001], [100/001/001], [011/010/000], [011/001/000], [010/010/010], [010/010/001], [010/001/001], [001/001/001].
%p A360660 g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
%p A360660       g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
%p A360660     end:
%p A360660 a:= n-> coeff(g(n$3), x, n):
%p A360660 seq(a(n), n=0..20);
%t A360660 g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i] + k - 1, k], {k, 0, j}]]]];
%t A360660 a[n_] := SeriesCoefficient[g[n, n, n], {x, 0, n}];
%t A360660 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 28 2023, after _Alois P. Heinz_ *)
%t A360660 Table[SeriesCoefficient[Product[1/(1 - x^k)^Binomial[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 15 2025 *)
%Y A360660 Main diagonal of A220886.
%Y A360660 Cf. A091058, A091059, A154323, A246107, A256070, A360664, A383073.
%K A360660 nonn
%O A360660 0,3
%A A360660 _Alois P. Heinz_, Feb 15 2023