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

0,3

Links

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

Formula

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

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 *)

Prog

(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

Crossrefs

Column k=3 of A293112.

Cf. A001006.

Sequence in context: A263734 A263735 A263731 * A299119 A348010 A271895

Adjacent sequences: A293739 A293740 A293741 * A293743 A293744 A293745

Keyword

nonn

Author

Alois P. Heinz, Oct 15 2017

Status

approved