cp's OEIS Frontend

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.

%I A340409 #7 Jan 06 2021 19:22:04
%S A340409 1,1,3,7,18,42,110,250,627,1439,3523,8063,19374,44274,104816,238976,
%T A340409 559171,1271295,2946901,6679741,15363719,34719631,79335385,178749829,
%U A340409 406164359,912475815,2063298409,4622461673,10407679805,23254807241,52160338735,116252939071
%N 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.
%H A340409 Alois P. Heinz, <a href="/A340409/b340409.txt">Table of n, a(n) for n = 0..1000</a>
%F A340409 G.f.: Product_{j>=1} (1+x^j)^A027306(j).
%e A340409 a(3) = 7: {aaa}, {aab}, {aba}, {baa}, {aa,a}, {ab,a}, {ba,a}.
%p A340409 b:= proc(n, i, t) option remember; `if`(t=1, 1/n!,
%p A340409       add(b(n-j, j, t-1)/j!, j=i..n/t))
%p A340409     end:
%p A340409 g:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
%p A340409 h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A340409       add(h(n-i*j, i-1, k)*binomial(g(i, k), j), j=0..n/i)))
%p A340409     end:
%p A340409 a:= n-> h(n$2, min(n, 2)):
%p A340409 seq(a(n), n=0..32);
%Y A340409 Column k=2 of A292795.
%Y A340409 Cf. A027306.
%K A340409 nonn
%O A340409 0,3
%A A340409 _Alois P. Heinz_, Jan 06 2021