cp's OEIS Frontend

A292839 Number of sets of nonempty words with a total of n letters over 5-ary alphabet.

%I A292839 #10 Dec 30 2020 08:22:33
%S A292839 1,5,35,260,1805,12376,83175,550775,3600400,23276175,149012380,
%T A292839 945726575,5955676150,37243117575,231412658225,1429522303905,
%U A292839 8783382129825,53700395135475,326809026132350,1980383108328950,11952682268739660,71870696616619250,430632502970026125
%N A292839 Number of sets of nonempty words with a total of n letters over 5-ary alphabet.
%H A292839 Alois P. Heinz, <a href="/A292839/b292839.txt">Table of n, a(n) for n = 0..1000</a>
%F A292839 G.f.: Product_{j>=1} (1+x^j)^(5^j).
%F A292839 a(n) ~ 5^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(5^(m-1)-1)) = 0.112852293193143374268678097722831649456... - _Vaclav Kotesovec_, Sep 28 2017
%p A292839 h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A292839       add(h(n-i*j, i-1)*binomial(5^i, j), j=0..n/i)))
%p A292839     end:
%p A292839 a:= n-> h(n$2):
%p A292839 seq(a(n), n=0..30);
%t A292839 h[n_, i_] := h[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A292839     Sum[h[n - i j, i - 1] Binomial[5^i, j], {j, 0, n/i}]]];
%t A292839 a[n_] := h[n, n];
%t A292839 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 30 2020, after _Alois P. Heinz_ *)
%Y A292839 Column k=5 of A292804.
%K A292839 nonn
%O A292839 0,2
%A A292839 _Alois P. Heinz_, Sep 24 2017