A293742 Number of sets 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, 2, 6, 14, 39, 104, 284, 775, 2145, 5941, 16563, 46329, 130100, 366432, 1035191, 2931797, 8323290, 23680142, 67505721, 192791938, 551537506, 1580315319, 4534715008, 13030197881, 37489497472, 107991978290, 311433926717, 899093131819, 2598257241179

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=3 of A293112.

Cf. A001006.

Programs

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:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..35);

Mathematica

With[{n = 29}, CoefficientList[Series[Product[(1 + x^j)^Hypergeometric2F1[(1 - j)/2, -j/2, 2, 4], {j, n}], {x, 0, n}], x]] (* Michael De Vlieger, Oct 15 2017, after Peter Luschny at A001006 *)

Python

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

Formula

G.f.: Product_{j>=1} (1+x^j)^A001006(j).