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

%I A252654 #23 Nov 21 2018 02:48:50
%S A252654 1,1,7,64,843,13876,276792,6438797,170938483,5091463423,167965714273,
%T A252654 6074571662270,238837895468954,10138497426332796,461941179848628434,
%U A252654 22478593443737857695,1163160397700757351363,63760710281671647692688,3690276585886363643056992
%N A252654 Number of multisets of nonempty words with a total of n letters over n-ary alphabet.
%H A252654 Alois P. Heinz, <a href="/A252654/b252654.txt">Table of n, a(n) for n = 0..350</a>
%F A252654 a(n) = [x^n] Product_{j>=1} 1/(1-x^j)^(n^j).
%F A252654 a(n) = n-th term of the Euler transform of the powers of n.
%F A252654 a(n) ~ n^(n-3/4) * exp(2*sqrt(n) - 1/2) / (2*sqrt(Pi)). - _Vaclav Kotesovec_, Mar 14 2015
%F A252654 a(n) = [x^n] exp(n*Sum_{k>=1} x^k/(k*(1 - n*x^k))). - _Ilya Gutkovskiy_, Nov 20 2018
%e A252654 a(2) = 7: {aa}, {ab}, {ba}, {bb}, {a,a}, {a,b}, {b,b}.
%p A252654 with(numtheory):
%p A252654 A:= proc(n, k) option remember; `if`(n=0, 1, add(add(
%p A252654        d*k^d, d=divisors(j)) *A(n-j, k), j=1..n)/n)
%p A252654     end:
%p A252654 a:= n-> A(n$2):
%p A252654 seq(a(n), n=0..25);
%t A252654 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];
%t A252654 a[n_] := A[n, n];
%t A252654 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Mar 21 2017, translated from Maple *)
%Y A252654 Main diagonal of A144074.
%Y A252654 Cf. A292805, A292873.
%K A252654 nonn
%O A252654 0,3
%A A252654 _Alois P. Heinz_, Dec 19 2014
%E A252654 New name from comment by _Alois P. Heinz_, Sep 21 2018