A210737 Number of Dyck n-paths all of whose ascents have prime lengths.

1, 0, 1, 1, 2, 6, 8, 29, 50, 141, 327, 771, 2047, 4746, 12644, 30941, 79886, 204885, 522242, 1365056, 3505825, 9185742, 23907116, 62636476, 164624803, 432540010, 1142827935, 3017208675, 7996379870, 21211540268, 56369770281, 150086840133, 400009010758

Offset: 0

Author

Alois P. Heinz, May 10 2012

Keywords

• nonn

Examples

			a(0) = 1: the empty path.
a(1) = 0.
a(2) = 1: UUDD.
a(3) = 1: UUUDDD.
a(4) = 2: UUDDUUDD, UUDUUDDD.
a(5) = 6: UUDDUUUDDD, UUDUUUDDDD, UUUDDDUUDD, UUUDDUUDDD, UUUDUUDDDD, UUUUUDDDDD.
a(6) = 8: UUDDUUDDUUDD, UUDDUUDUUDDD, UUDUUDDDUUDD, UUDUUDDUUDDD, UUDUUDUUDDDD, UUUDDDUUUDDD, UUUDDUUUDDDD, UUUDUUUDDDDD.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..700

Crossrefs

Cf. A210735.

Programs

Maple

with(numtheory):
b:= proc(x, y, u) option remember;
      `if`(x<0 or  y b(n, n, true):
seq(a(n), n=0..40);

Mathematica

b[x_, y_, u_] := b[x, y, u] = If[x<0 || yJean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

PARI

seq(n)={Vec(serreverse(x/(1 + sum(i=2, n, if(isprime(i), x^i))) + O(x*x^n)))} \\ Andrew Howroyd, Apr 28 2018

Formula

a(n) ~ c * d^n / n^(3/2), where d = 2.7925684676903082567..., c = 0.4016264581712556... . - Vaclav Kotesovec, Sep 02 2014