A293965 Number of sets of exactly three nonempty words with a total of n letters over n-ary 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, 8, 30, 114, 411, 1462, 5237, 18998, 70220, 265010, 1024692, 4059100, 16504058, 68843340, 294854550, 1295771712, 5843980456, 27026394156, 128135282356, 622230803212, 3093321051636, 15728089431744, 81739630155456, 433801710925696, 2349410730317456

Offset: 5

Author

Alois P. Heinz, Oct 20 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 5..803

Crossrefs

Column k=3 of A293815.

Cf. A000085.

Programs

Maple

g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*binomial(g(i), j)*x^j, j=0..n/i))), x, 4)
    end:
a:= n-> coeff(b(n$2), x, 3):
seq(a(n), n=5..30);

Formula

a(n) = [x^n y^3] Product_{j>=1} (1+y*x^j)^A000085(j).