A212386 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 6).

1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1718, 3454, 6646, 12841, 26589, 61813, 158918, 426401, 1134431, 2914055, 7171539, 16967745, 39008002, 88529366, 202057561, 471422866, 1133448790, 2799775102, 7026467132, 17684574313, 44192085565, 109081884957

Offset: 0

Author

Alois P. Heinz, May 12 2012

Keywords

• nonn

Comments

Lengths of descents are unrestricted.

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Asymptotic of subsequences of A212382

Crossrefs

Column k=6 of A212382.

Programs

Maple

b:= proc(x, y, u) option remember;
      `if`(x<0 or  y 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 *)

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 */

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