A212384 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 4).

1, 1, 1, 1, 1, 2, 7, 22, 57, 128, 268, 573, 1343, 3434, 9038, 23374, 58649, 144400, 355992, 892336, 2280020, 5892301, 15253305, 39347067, 101177783, 260255812, 671941182, 1743500452, 4542147622, 11858732144, 30983904244, 80982376879, 211831943129, 554905957520

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 = 4*(1-2*s+s^2)/(s*(4*s-3)) = 0.36467312501521477251..., where s = A(r) is described below. - Vaclav Kotesovec, Mar 21 2014

Examples

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(5) = 2: UDUDUDUDUD, UUUUUDDDDD.
a(6) = 7: UDUDUDUDUDUD, UDUUUUUDDDDD, UUUUUDDDDDUD, UUUUUDDDDUDD, UUUUUDDDUDDD, UUUUUDDUDDDD, UUUUUDUDDDDD.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Recurrence (of order 8)
• Vaclav Kotesovec, Asymptotic of subsequences of A212382

Crossrefs

Column k=4 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)^4), A), x, n+1), x, n):
seq(a(n), n=0..40);

Mathematica

a[n_] := Sum[Binomial[3k-2n-1, n-k]*Binomial[n+1, 4k-3n], {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(3*k-2*n-1,n-k)*binomial(n+1,4*k-3*n),k,0,n)/(n+1); /* Vladimir Kruchinin, Mar 05 2016 */

PARI

a(n) = sum(k=0, n, binomial(3*k-2*n-1,n-k)*binomial(n+1,4*k-3*n))/(n+1); \\ Michel Marcus, Mar 05 2016

Formula

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^4).

a(n) ~ (s*(4*s-3))^(n+3/2) / (sqrt(Pi) * sqrt(5*s-3) * n^(3/2) * 2^(2*n+9/2) * (s-1)^(2*n+7/2)), where s = 1.880470225526517115847397... is the root of the equation 283 - 2156*s + 7312*s^2 - 14400*s^3 + 17920*s^4 - 14336*s^5 + 7168*s^6 - 2048*s^7 + 256*s^8 = 0. - Vaclav Kotesovec, Mar 21 2014

a(n) = Sum_{k=0..n} (binomial(3*k-2*n-1,n-k)*binomial(n+1,4*k-3*n))/(n+1). - Vladimir Kruchinin, Mar 05 2016