A212385 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 5).

1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 464, 943, 1873, 3914, 9101, 23298, 61915, 162283, 409888, 996456, 2360486, 5555333, 13244114, 32357022, 80958851, 205389082, 522000262, 1317987172, 3297123652, 8190326857, 20302864970, 50482613327, 126318440989

Offset: 0

Author

Alois P. Heinz, May 12 2012

Keywords

• nonn

Comments

Lengths of descents are unrestricted.

The radius of convergence of g.f. A(x) is r = 5*(1-2*s+s^2)/(s*(5*s-4)) = 0.3804593157188..., where s = A(r) is described below. - Vaclav Kotesovec, Mar 20 2014

Examples

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(6) = 2: UDUDUDUDUDUD, UUUUUUDDDDDD.
a(7) = 8: UDUDUDUDUDUDUD, UDUUUUUUDDDDDD, UUUUUUDDDDDDUD, UUUUUUDDDDDUDD, UUUUUUDDDDUDDD, UUUUUUDDDUDDDD, UUUUUUDDUDDDDD, UUUUUUDUDDDDDD.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Recurrence (of order 10)
• Vaclav Kotesovec, Asymptotic of subsequences of A212382

Crossrefs

Column k=5 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)^5), A), x, n+1), x, n):
seq(a(n), n=0..40);

Mathematica

b[x_, y_, u_] := b[x, y, u] = If[x<0 || yJean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Maxima

a(n):=sum(binomial(4*k-3*n-1, n-k)*binomial(n+1, 5*k-4*n), k, 0, n)/(n+1); /* Vladimir Kruchinin, Mar 05 2016 */

PARI

a(n) = sum(k=0, n, binomial(4*k-3*n-1,n-k)*binomial(n+1,5*k-4*n))/(n+1); \\ Michel Marcus, Mar 05 2016

Formula

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^5).

Representation in terms of special values of generalized hypergeometric function of type 12F11: a(n) = hypergeom([1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -(1/6)*n, -(1/6)*n+5/6, -(1/6)*n+2/3, -(1/6)*n+1/2, -(1/6)*n+1/3, 1/6-(1/6)*n], [1/6, 1/3, 1/3, 1/2, 1/2, 2/3, 2/3, 5/6, 5/6, 1, 7/6], 7^7/6^6), n>=0. - Karol A. Penson, Jun 21 2013

a(n) ~ s^(n+3/2) * (5*s-4)^(n+2) / (2 * sqrt(Pi) * sqrt(3*s-2) * n^(3/2) * 5^(n+5/2) * (s-1)^(2*n+9/2)), where s = 1.87696911628429... is the root of the equation 2869 - 29970*s + 138225*s^2 - 373000*s^3 + 655625*s^4 - 787500*s^5 + 656250*s^6 - 375000*s^7 + 140625*s^8 - 31250*s^9 + 3125*s^10 = 0. - Vaclav Kotesovec, Mar 20 2014

a(n) = Sum_{k=0..n}(binomial(4*k-3*n-1,n-k)*binomial(n+1,5*k-4*n))/(n+1). - Vladimir Kruchinin, Mar 05 2016