A293733 Number of multisets of nonempty words with a total of n letters over ternary 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, 19, 48, 131, 348, 954, 2607, 7212, 19995, 55816, 156246, 439267, 1238397, 3502004, 9927260, 28208628, 80322048, 229161413, 654966245, 1875074366, 5376298225, 15437286706, 44385247519, 127776425727, 368276055467, 1062618076382, 3069264747076

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)^A001006(j).

a(n) ~ c * 3^n / n^(3/2), where c = 11.84175157992103588081767200532703865225243959779980786519467770732598276486... - Vaclav Kotesovec, May 30 2019

Maple

g:= proc(n) option remember; `if`(n<2, 1,
      g(n-1)+add(g(k)*g(n-k-2), k=0..n-2))
    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, g[n-1] + Sum[g[k]*g[n-k-2], {k, 0, n-2}]];
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 g(n - 1) + sum([g(k)*g(n - k - 2) for k in range(n - 1)])
@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(map(a, range(36))) # Indranil Ghosh, Oct 15 2017

Crossrefs

Column k=3 of A293108.

Cf. A001006.

Sequence in context: A049117 A146810 A246493 * A073063 A370169 A007288

Adjacent sequences: A293730 A293731 A293732 * A293734 A293735 A293736

Keyword

nonn

Author

Alois P. Heinz, Oct 15 2017

Status

approved