A252654 Number of multisets of nonempty words with a total of n letters over n-ary alphabet.

1, 1, 7, 64, 843, 13876, 276792, 6438797, 170938483, 5091463423, 167965714273, 6074571662270, 238837895468954, 10138497426332796, 461941179848628434, 22478593443737857695, 1163160397700757351363, 63760710281671647692688, 3690276585886363643056992

Offset: 0

Author

Alois P. Heinz, Dec 19 2014

Keywords

• nonn

Examples

			a(2) = 7: {aa}, {ab}, {ba}, {bb}, {a,a}, {a,b}, {b,b}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350

Crossrefs

Main diagonal of A144074.

Cf. A292805, A292873.

Programs

Maple

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
       d*k^d, d=divisors(j)) *A(n-j, k), j=1..n)/n)
    end:
a:= n-> A(n$2):
seq(a(n), n=0..25);

Mathematica

A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[d*k^d, {d, Divisors[j]}]*A[n - j, k], {j, 1, n}]/n];
a[n_] := A[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

Formula

a(n) = [x^n] Product_{j>=1} 1/(1-x^j)^(n^j).

a(n) = n-th term of the Euler transform of the powers of n.

a(n) ~ n^(n-3/4) * exp(2*sqrt(n) - 1/2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Mar 14 2015

a(n) = [x^n] exp(n*Sum_{k>=1} x^k/(k*(1 - n*x^k))). - Ilya Gutkovskiy, Nov 20 2018

Extensions

New name from comment by Alois P. Heinz, Sep 21 2018