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%I #14 Dec 19 2020 02:29:28
%S 1,1,3,7,20,54,163,492,1571,5122,17262,59483,209958,755615,2770994,
%T 10330036,39103166,150073289,583329574,2293822828,9116935874,
%U 36593731182,148221246775,605427601519,2492286544749,10334197803358,43140208034891,181224681022614
%N Number of multisets of nonempty words with a total of n letters over quinary 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=6.
%H Alois P. Heinz, <a href="/A293735/b293735.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: Product_{j>=1} 1/(1-x^j)^A049401(j).
%F a(n) ~ c * 5^n / n^5, where c = 542.824729617782144... - _Vaclav Kotesovec_, May 30 2019
%p g:= proc(n) option remember;
%p `if`(n<3, [1, 1, 2][n+1], ((3*n^2+17*n+15)*g(n-1)
%p +(n-1)*(13*n+9)*g(n-2) -15*(n-1)*(n-2)*g(n-3)) /
%p ((n+4)*(n+6)))
%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);
%t g[n_] := g[n] = If[n < 3, {1, 1, 2}[[n+1]], ((3n^2+17n+15) g[n-1] + (n-1)(13n+9) g[n-2] - 15(n-1)(n-2) g[n-3]) / ((n+4)(n+6))];
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[g[d] d, {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
%t a /@ Range[0, 35] (* _Jean-François Alcover_, Dec 19 2020, after _Alois P. Heinz_ *)
%Y Column k=5 of A293108.
%Y Cf. A049401, A293110, A293744.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Oct 15 2017
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