A293732 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.
1, 1, 3, 6, 15, 31, 73, 155, 351, 755, 1673, 3604, 7897, 16988, 36902, 79222, 171030, 366180, 786746, 1679976, 3595207, 7657631, 16332935, 34706319, 73812099, 156503351, 332004423, 702533059, 1486998780, 3140716766, 6634315264, 13988517803, 29494816751
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..3260 (terms 0..1000 from Alois P. Heinz)
Programs
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Maple
a:= proc(n) option remember; `if`(n=0, 1, add(add(binomial(d, floor(d/2))*d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n) end: seq(a(n), n=0..35); -
Mathematica
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 *)
Formula
G.f.: Product_{j>=1} 1/(1-x^j)^A001405(j).
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