A292838 Number of sets of nonempty words with a total of n letters over quaternary alphabet.

1, 4, 22, 132, 729, 4000, 21488, 113760, 594548, 3073392, 15732936, 79846448, 402104884, 2010879968, 9992425872, 49366096352, 242584319710, 1186177166680, 5773569726884, 27982357252632, 135079969593838, 649640609539360, 3113354757088720, 14871179093155424

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=4 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(4^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[4^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)^(4^j).

a(n) ~ 4^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(4^(m-1)-1)) = 0.147762663788961720137665013823002812172... - Vaclav Kotesovec, Sep 28 2017