OFFSET
0,11
COMMENTS
Lengths of descents are unrestricted.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..800
Vaclav Kotesovec, Asymptotic of subsequences of A212382
FORMULA
G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^9).
a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where p = 9 and r = 0.4164039515514120671..., s = 1.882616423435763466... are roots of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014
EXAMPLE
a(0) = 1: the empty path.
a(1) = 1: UD.
a(10) = 2: UDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUDDDDDDDDDD.
a(11) = 12: UDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUDDDDDDDDDD, UUUUUUUUUUDDDDDDDDDDUD, UUUUUUUUUUDDDDDDDDDUDD, UUUUUUUUUUDDDDDDDDUDDD, UUUUUUUUUUDDDDDDDUDDDD, UUUUUUUUUUDDDDDDUDDDDD, UUUUUUUUUUDDDDDUDDDDDD, UUUUUUUUUUDDDDUDDDDDDD, UUUUUUUUUUDDDUDDDDDDDD, UUUUUUUUUUDDUDDDDDDDDD, UUUUUUUUUUDUDDDDDDDDDD.
MAPLE
b:= proc(x, y, u) option remember;
`if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
`if`(u, add (b(x-(9*t+1), y, false), t=0..(x-1)/9), 0)))
end:
a:= n-> b(n$2, true):
seq(a(n), n=0..40);
# second Maple program:
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^9), A), x, n+1), x, n):
seq(a(n), n=0..40);
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 12 2012
STATUS
approved