OFFSET
0,8
COMMENTS
Lengths of descents are unrestricted.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotic of subsequences of A212382
FORMULA
G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^6).
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 = 6 and r = 0.3925132712580446244..., s = 1.876653786643058101... 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
a(n) = Sum_{k=0..n} (binomial(5*k-4*n-1,n-k)*binomial(n+1,6*k-5*n))/(n+1). - Vladimir Kruchinin, Mar 05 2016
EXAMPLE
a(0) = 1: the empty path.
a(1) = 1: UD.
a(7) = 2: UDUDUDUDUDUDUD, UUUUUUUDDDDDDD.
a(8) = 9: UDUDUDUDUDUDUDUD, UDUUUUUUUDDDDDDD, UUUUUUUDDDDDDDUD, UUUUUUUDDDDDDUDD, UUUUUUUDDDDDUDDD, UUUUUUUDDDDUDDDD, UUUUUUUDDDUDDDDD, UUUUUUUDDUDDDDDD, UUUUUUUDUDDDDDDD.
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-(6*t+1), y, false), t=0..(x-1)/6), 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)^6), A), x, n+1), x, n):
seq(a(n), n=0..40);
MATHEMATICA
a[n_] := Sum[Binomial[5k-4n-1, n-k]*Binomial[n+1, 6k-5n], {k, 0, n}]/(n+1);
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 03 2017, after Vladimir Kruchinin *)
PROG
(Maxima)
a(n):=sum(binomial(5*k-4*n-1, n-k)*binomial(n+1, 6*k-5*n), k, 0, n)/(n+1); /* Vladimir Kruchinin, Mar 05 2016 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 12 2012
STATUS
approved