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

0,3

Links

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

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

Example

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

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 *)

Crossrefs

Main diagonal of A144074.

Cf. A292805, A292873.

Sequence in context: A173516 A197785 A360465 * A352841 A197787 A024095

Adjacent sequences: A252651 A252652 A252653 * A252655 A252656 A252657

Keyword

nonn

Author

Alois P. Heinz, Dec 19 2014

Extensions

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

Status

approved