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

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..3260 (terms 0..1000 from Alois P. Heinz)

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

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 *)

Crossrefs

Column k=2 of A293108.

Cf. A001405.

Sequence in context: A244706 A097941 A317792 * A117161 A319765 A244707

Adjacent sequences: A293729 A293730 A293731 * A293733 A293734 A293735

Keyword

nonn

Author

Alois P. Heinz, Oct 15 2017

Status

approved