A293734 Number of multisets of nonempty words with a total of n letters over quaternary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

1, 1, 3, 7, 20, 53, 157, 455, 1393, 4270, 13495, 42907, 139323, 455182, 1510831, 5042858, 17044789, 57891598, 198665585, 684615958, 2379765470, 8302157207, 29177909254, 102867895209, 364981305292, 1298526198294, 4645569147108, 16659856695779, 60036951331540

Offset: 0

Author

Alois P. Heinz, Oct 15 2017

Keywords

• nonn

Links

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

Crossrefs

Column k=4 of A293108.

Cf. A005817.

Programs

Maple

g:= proc(n) option remember; `if`(n<2, 1, (4*(2*n+3)*
       g(n-1)+16*(n-1)*n*g(n-2))/((n+3)*(n+4)))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(g(d)
      *d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

Mathematica

g[n_] := g[n] = If[n<2, 1, (4*(2*n+3)*g[n-1] + 16*(n-1)*n*g[n-2])/((n+3)* (n+4))];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[g[d]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 30 2019, after Alois P. Heinz *)

Python

from sympy.core.cache import cacheit
from sympy import divisors
@cacheit
def g(n): return 1 if n<2 else (4*(2*n + 3)*g(n - 1) + 16*(n - 1)*n*g(n - 2))//((n + 3)*(n + 4))
@cacheit
def a(n): return 1 if n==0 else sum(sum(g(d)*d for d in divisors(j))*a(n - j) for j in range(1, n + 1))//n
print([a(n) for n in range(36)]) # Indranil Ghosh, Oct 15 2017

Formula

G.f.: Product_{j>=1} 1/(1-x^j)^A005817(j).

a(n) ~ c * 4^n / n^3, where c = 19.002514794... - Vaclav Kotesovec, May 30 2019