%I #14 May 31 2019 02:38:08

%S 1,1,3,6,15,31,73,155,351,755,1673,3604,7897,16988,36902,79222,171030,

%T 366180,786746,1679976,3595207,7657631,16332935,34706319,73812099,

%U 156503351,332004423,702533059,1486998780,3140716766,6634315264,13988517803,29494816751

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

%H Vaclav Kotesovec, <a href="/A293732/b293732.txt">Table of n, a(n) for n = 0..3260</a> (terms 0..1000 from Alois P. Heinz)

%F G.f.: Product_{j>=1} 1/(1-x^j)^A001405(j).

%F a(n) ~ 2^(n - 1/6) * exp(3*(n/2)^(1/3) - 2 + S) / (sqrt(3*Pi) * n^(5/6)), where S = Sum_{k>=2} (sqrt(1/(1 - 1/2^(2*k - 2))) - 1) * (2^k + 2) / (2*k) = 0.3158684977247920135402311766405977266170498097655... - _Vaclav Kotesovec_, May 30 2019

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(binomial(d,

%p floor(d/2))*d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)

%p end:

%p seq(a(n), n=0..35);

%t nmax = 40; A001405 = Table[Binomial[n, Floor[n/2]], {n, 1, nmax}]; CoefficientList[Series[Product[1/(1 - x^k)^A001405[[k]], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, May 30 2019 *)

%Y Column k=2 of A293108.

%Y Cf. A001405.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Oct 15 2017