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

0,2

Links

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

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

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 *)

Crossrefs

Column k=4 of A292804.

Sequence in context: A180899 A007195 A356283 * A193620 A321275 A274745

Adjacent sequences: A292835 A292836 A292837 * A292839 A292840 A292841

Keyword

nonn

Author

Alois P. Heinz, Sep 24 2017

Status

approved