A360638 Number of sets of nonempty words over binary alphabet where each letter occurs n times.

1, 3, 16, 100, 593, 3497, 20316, 116378, 658214, 3679450, 20350028, 111459648, 605060633, 3257784589, 17408647968, 92378535290, 487031130699, 2552197485757, 13298890952222, 68930923717598, 355507581655752, 1824924721216084, 9326440815314046, 47464093855706540

Offset: 0

Author

Alois P. Heinz, Feb 14 2023

Keywords

• nonn

Examples

			a(0) = 1: {}.
a(1) = 3: {ab}, {ba}, {a,b}.
a(2) = 16: {aabb}, {abab}, {abba}, {baab}, {baba}, {bbaa}, {a,abb}, {a,bab}, {a,bba}, {aa,bb}, {aab,b}, {ab,ba}, {aba,b}, {b,baa}, {a,ab,b}, {a,b,ba}.
		

Links

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

Crossrefs

Cf. A080277, A360626 (the same for multisets), A360634.

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), 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], {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, Dec 09 2023, after Alois P. Heinz *)

Formula

a(n) = A360634(2n,n).

a(n) mod 2 = 1 <=> n in { A080277 } U {0}.