A293735 - OEIS

%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