A292805 Number of sets of nonempty words with a total of n letters over n-ary alphabet.

1, 1, 5, 55, 729, 12376, 250735, 5904746, 158210353, 4747112731, 157545928646, 5726207734545, 226093266070501, 9632339536696943, 440262935648935344, 21482974431740480311, 1114363790702406540897, 61219233429920494716931, 3550130647865299090804375

Offset: 0

Author

Alois P. Heinz, Sep 23 2017

Keywords

• nonn

Examples

			a(0) = 1: {}.
a(1) = 1: {a}.
a(2) = 5: {aa}, {ab}, {ba}, {bb}, {a,b}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..381

Crossrefs

Main diagonal of A292804.

Cf. A252654, A292845.

Programs

Maple

h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
    end:
a:= n-> h(n$3):
seq(a(n), n=0..20);

Mathematica

h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[h[n - i*j, i - 1, k]*Binomial[k^i, j], {j, 0, n/i}]]];
a[n_] := h[n, n, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 04 2018, from Maple *)

Formula

a(n) = [x^n] Product_{j=1..n} (1+x^j)^(n^j).

a(n) ~ n^(n - 3/4) * exp(2*sqrt(n) - 1/2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Aug 26 2019