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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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).

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);

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(map(a, 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

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Last modified April 20 17:54 EDT 2019. Contains 322310 sequences. (Running on oeis4.)