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A293734
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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.
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4
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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
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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FORMULA
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G.f.: Product_{j>=1} 1/(1-x^j)^A005817(j).
a(n) ~ c * 4^n / n^3, where c = 19.002514794... - Vaclav Kotesovec, May 30 2019
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MAPLE
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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);
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MATHEMATICA
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g[n_] := g[n] = If[n<2, 1, (4*(2*n+3)*g[n-1] + 16*(n-1)*n*g[n-2])/((n+3)* (n+4))];
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 *)
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PROG
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(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([a(n) for n in range(36)]) # Indranil Ghosh, Oct 15 2017
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CROSSREFS
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Column k=4 of A293108.
Cf. A005817.
Sequence in context: A058499 A003097 A109220 * A018034 A293735 A293736
Adjacent sequences: A293731 A293732 A293733 * A293735 A293736 A293737
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KEYWORD
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nonn
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AUTHOR
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Alois P. Heinz, Oct 15 2017
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STATUS
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approved
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