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%I #9 Dec 19 2020 03:08:31
%S 1,1,3,7,20,54,164,500,1630,5471,19246,70020,264961,1035540,4187725,
%T 17440159,74817905,329400093,1487844185,6873585346,32460719143,
%U 156315314070,767106102127,3828629444020,19423438144438,99998608025751,522200287437179,2762351298913471
%N Number of multisets of nonempty words with a total of n letters over octonary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
%C This sequence differs from A293110 first at n=9.
%H Alois P. Heinz, <a href="/A293738/b293738.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: Product_{j>=1} 1/(1-x^j)^A007580(j).
%F a(n) ~ c * 8^n / n^14, where c = 4485962145436.6348123684794... - _Vaclav Kotesovec_, Dec 19 2020
%p g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],
%p ((40*n^3+1084*n^2+8684*n+18480)*g(n-1) +16*(n-1)*
%p (5*n^3+107*n^2+610*n+600)*g(n-2) -1024*(n-1)*(n-2)*
%p (n+6)*g(n-3) -1024*(n-1)*(n-2)*(n-3)*(n+4)*g(n-4))
%p /((n+7)*(n+12)*(n+15)*(n+16)))
%p end:
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(g(d)
%p *d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..35);
%Y Column k=8 of A293108.
%Y Cf. A007580, A293110, A293747.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Oct 15 2017
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