A293735 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.

1, 1, 3, 7, 20, 54, 163, 492, 1571, 5122, 17262, 59483, 209958, 755615, 2770994, 10330036, 39103166, 150073289, 583329574, 2293822828, 9116935874, 36593731182, 148221246775, 605427601519, 2492286544749, 10334197803358, 43140208034891, 181224681022614

Offset: 0

Author

Alois P. Heinz, Oct 15 2017

Keywords

• nonn

Comments

This sequence differs from A293110 first at n=6.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=5 of A293108.

Cf. A049401, A293110, A293744.

Programs

Maple

g:= proc(n) option remember;
      `if`(n<3, [1, 1, 2][n+1], ((3*n^2+17*n+15)*g(n-1)
       +(n-1)*(13*n+9)*g(n-2) -15*(n-1)*(n-2)*g(n-3)) /
       ((n+4)*(n+6)))
    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 < 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))];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[g[d] d, {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
a /@ Range[0, 35] (* Jean-François Alcover, Dec 19 2020, after Alois P. Heinz *)

Formula

G.f.: Product_{j>=1} 1/(1-x^j)^A049401(j).

a(n) ~ c * 5^n / n^5, where c = 542.824729617782144... - Vaclav Kotesovec, May 30 2019