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

0,8

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

Column k=6 of A212382.

Sequence in context: A306721 A306852 A275425 * A333883 A206374 A037553

Adjacent sequences: A212383 A212384 A212385 * A212387 A212388 A212389

Keyword

nonn

Author

Alois P. Heinz, May 12 2012

Status

approved