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

0,3

Links

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

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

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 *)

Prog

(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

Crossrefs

Column k=4 of A293108.

Cf. A005817.

Sequence in context: A058499 A003097 A109220 * A018034 A293735 A293736

Adjacent sequences: A293731 A293732 A293733 * A293735 A293736 A293737

Keyword

nonn

Author

Alois P. Heinz, Oct 15 2017

Status

approved