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A360626 Number of multisets of nonempty words over binary alphabet where each letter occurs n times.

%I A360626 #18 Nov 17 2023 11:35:14
%S A360626 1,3,21,131,830,5066,30456,179256,1038593,5928071,33402561,186021335,
%T A360626 1025162709,5596047683,30282832593,162573152651,866385400935,
%U A360626 4585861723905,24120596727003,126124094912499,655868112470175,3393060517486981,17468543071082489
%N A360626 Number of multisets of nonempty words over binary alphabet where each letter occurs n times.
%H A360626 Alois P. Heinz, <a href="/A360626/b360626.txt">Table of n, a(n) for n = 0..380</a>
%F A360626 a(n) = [x^(2n)*y^n] Product_{i>=1} Product_{j=0..i} 1/(1-x^i*y^j)^binomial(i,j).
%F A360626 a(n) = A055375(2n,n).
%e A360626 a(0) = 1: {}.
%e A360626 a(1) = 3: {ab}, {ba}, {a,b}.
%e A360626 a(2) = 21: {aabb}, {abab}, {abba}, {baab}, {baba}, {bbaa}, {a,abb}, {a,bab}, {a,bba}, {aa,bb}, {aab,b}, {ab,ab}, {ab,ba}, {aba,b}, {b,baa}, {ba,ba}, {a,a,bb}, {a,ab,b}, {a,b,ba}, {aa,b,b}, {a,a,b,b}.
%p A360626 g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
%p A360626       g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
%p A360626     end:
%p A360626 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A360626      `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
%p A360626     end:
%p A360626 a:= n-> coeff(b(2*n$2), x, n):
%p A360626 seq(a(n), n=0..31);
%t A360626 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 A360626 b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
%t A360626 a[n_] := Coefficient[b[2n, 2n], x, n];
%t A360626 Table[a[n], {n, 0, 31}] (* _Jean-François Alcover_, Nov 17 2023, after _Alois P. Heinz_ *)
%Y A360626 Cf. A055375, A359962, A360638 (the same for sets).
%K A360626 nonn
%O A360626 0,2
%A A360626 _Alois P. Heinz_, Feb 14 2023