A293743 Number of sets 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, 2, 6, 15, 44, 129, 386, 1185, 3690, 11725, 37578, 122577, 402477, 1340640, 4490368, 15219148, 51825464, 178235039, 615461671, 2143127872, 7488890027, 26357539204, 93050275129, 330544091758, 1177338456789, 4216288462832, 15134924595039, 54588972553934

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)^A005817(j).

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:
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

g[n_] := g[n] = If[n<2, 1, (4(2n+3) g[n-1]+16(n-1) n g[n-2])/((n+3)(n+4))];
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[b[n-i j, i-1] Binomial[ g[i], j], {j, 0, n/i}]]];
a[n_] := b[n, n];
Array[a, 35, 0] (* Jean-François Alcover, May 31 2019, from Maple *)

Prog

(Python)
from sympy.core.cache import cacheit
from sympy import binomial
@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 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=4 of A293112.

Cf. A005817.

Sequence in context: A151515 A264746 A052870 * A360274 A001444 A293744

Adjacent sequences: A293740 A293741 A293742 * A293744 A293745 A293746

Keyword

nonn

Author

Alois P. Heinz, Oct 15 2017

Status

approved