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

1, 3, 21, 131, 830, 5066, 30456, 179256, 1038593, 5928071, 33402561, 186021335, 1025162709, 5596047683, 30282832593, 162573152651, 866385400935, 4585861723905, 24120596727003, 126124094912499, 655868112470175, 3393060517486981, 17468543071082489

Offset: 0

Author

Alois P. Heinz, Feb 14 2023

Keywords

• nonn

Examples

			a(0) = 1: {}.
a(1) = 3: {ab}, {ba}, {a,b}.
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}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..380

Crossrefs

Cf. A055375, A359962, A360638 (the same for sets).

Programs

Maple

g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
a:= n-> coeff(b(2*n$2), x, n):
seq(a(n), n=0..31);

Mathematica

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}]]]];
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}]]]];
a[n_] := Coefficient[b[2n, 2n], x, n];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *)

Formula

a(n) = [x^(2n)*y^n] Product_{i>=1} Product_{j=0..i} 1/(1-x^i*y^j)^binomial(i,j).

a(n) = A055375(2n,n).