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

0,3

Comments

This sequence differs from A293110 first at n=6.

Links

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

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

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 *)

Crossrefs

Column k=5 of A293108.

Cf. A049401, A293110, A293744.

Sequence in context: A109220 A293734 A018034 * A293736 A293737 A293738

Adjacent sequences: A293732 A293733 A293734 * A293736 A293737 A293738

Keyword

nonn

Author

Alois P. Heinz, Oct 15 2017

Status

approved