A340409 Number of sets of nonempty words with a total of n letters over binary alphabet such that within each word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

1, 1, 3, 7, 18, 42, 110, 250, 627, 1439, 3523, 8063, 19374, 44274, 104816, 238976, 559171, 1271295, 2946901, 6679741, 15363719, 34719631, 79335385, 178749829, 406164359, 912475815, 2063298409, 4622461673, 10407679805, 23254807241, 52160338735, 116252939071

Offset

0,3

Links

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

Formula

G.f.: Product_{j>=1} (1+x^j)^A027306(j).

Example

a(3) = 7: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}.

Maple

b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
      add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))
    end:
a:= n-> h(n$2, min(n, 2)):
seq(a(n), n=0..32);

Crossrefs

Column k=2 of A292795.

Cf. A027306.

Sequence in context: A319001 A036669 A000633 * A374640 A091621 A129921

Adjacent sequences: A340406 A340407 A340408 * A340410 A340411 A340412

Keyword

nonn

Author

Alois P. Heinz, Jan 06 2021

Status

approved