A294005 Number of multisets 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, 2, 7, 22, 68, 218, 721, 2438, 8491, 30478, 112524, 428382, 1678600, 6778708, 28169286, 120516092, 530081370, 2396797920, 11125584584, 52993063796, 258676491628, 1293160049244, 6612750833996, 34564483264256, 184470133103464, 1004514566402816

Offset: 3

Author

Alois P. Heinz, Oct 21 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..802

Crossrefs

Column k=3 of A293808.

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 or i=1, x^n,
      add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 4)
    end:
a:= n-> coeff(b(n$2), x, 3):
seq(a(n), n=3..30);

Formula

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