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

1, 5, 35, 260, 1805, 12376, 83175, 550775, 3600400, 23276175, 149012380, 945726575, 5955676150, 37243117575, 231412658225, 1429522303905, 8783382129825, 53700395135475, 326809026132350, 1980383108328950, 11952682268739660, 71870696616619250, 430632502970026125

Offset: 0

Author

Alois P. Heinz, Sep 24 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=5 of A292804.

Programs

Maple

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(5^i, j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..30);

Mathematica

h[n_, i_] := h[n, i] = If[n == 0, 1, If[i < 1, 0,
    Sum[h[n - i j, i - 1] Binomial[5^i, j], {j, 0, n/i}]]];
a[n_] := h[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 30 2020, after Alois P. Heinz *)

Formula

G.f.: Product_{j>=1} (1+x^j)^(5^j).

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