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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). a(n) ~ c * 4^n / n^3, where c = 19.002514794... - Vaclav Kotesovec, May 30 2019 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); MATHEMATICA 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 *) 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([a(n) for n in 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 March 24 06:15 EDT 2023. Contains 361454 sequences. (Running on oeis4.)