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

%I A292805 #16 Aug 26 2019 04:54:03
%S A292805 1,1,5,55,729,12376,250735,5904746,158210353,4747112731,157545928646,
%T A292805 5726207734545,226093266070501,9632339536696943,440262935648935344,
%U A292805 21482974431740480311,1114363790702406540897,61219233429920494716931,3550130647865299090804375
%N A292805 Number of sets of nonempty words with a total of n letters over n-ary alphabet.
%H A292805 Alois P. Heinz, <a href="/A292805/b292805.txt">Table of n, a(n) for n = 0..381</a>
%F A292805 a(n) = [x^n] Product_{j=1..n} (1+x^j)^(n^j).
%F A292805 a(n) ~ n^(n - 3/4) * exp(2*sqrt(n) - 1/2) / (2*sqrt(Pi)). - _Vaclav Kotesovec_, Aug 26 2019
%e A292805 a(0) = 1: {}.
%e A292805 a(1) = 1: {a}.
%e A292805 a(2) = 5: {aa}, {ab}, {ba}, {bb}, {a,b}.
%p A292805 h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A292805       add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
%p A292805     end:
%p A292805 a:= n-> h(n$3):
%p A292805 seq(a(n), n=0..20);
%t A292805 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}]]];
%t A292805 a[n_] := h[n, n, n];
%t A292805 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 04 2018, from Maple *)
%Y A292805 Main diagonal of A292804.
%Y A292805 Cf. A252654, A292845.
%K A292805 nonn
%O A292805 0,3
%A A292805 _Alois P. Heinz_, Sep 23 2017