|
|
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.
|
|
5
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|