A210735 Number of Dyck n-paths all of whose ascents and descents have prime lengths.

1, 0, 1, 1, 1, 4, 2, 10, 10, 22, 46, 64, 167, 245, 560, 1035, 1978, 4210, 7715, 16497, 31929, 65216, 133295, 266244, 553750, 1116404, 2308931, 4738660, 9742795, 20204902, 41622910, 86539105, 179358694, 373018581, 777157221, 1618773690, 3382590684, 7065505631

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) = 1: UUDDUUDD.
a(5) = 4: UUDDUUUDDD, UUUDDDUUDD, UUUDDUUDDD, UUUUUDDDDD.
a(6) = 2: UUDDUUDDUUDD, UUUDDDUUUDDD.
a(7) = 10: UUDDUUDDUUUDDD, UUDDUUUDDDUUDD, UUDDUUUDDUUDDD, UUDDUUUUUDDDDD, UUUDDDUUDDUUDD, UUUDDUUDDDUUDD, UUUDDUUDDUUDDD, UUUUUDDDDDUUDD, UUUUUDDUUDDDDD, UUUUUUUDDDDDDD.
a(8) = 10: UUDDUUDDUUDDUUDD, UUDDUUUDDDUUUDDD, UUUDDDUUDDUUUDDD, UUUDDDUUUDDDUUDD, UUUDDDUUUDDUUDDD, UUUDDDUUUUUDDDDD, UUUDDUUDDDUUUDDD, UUUDDUUUDDDUUDDD, UUUUUDDDDDUUUDDD, UUUUUDDDUUUDDDDD.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..900

Crossrefs

Cf. A210737.

Programs

Maple

with(numtheory):
b:= proc(x, y, u) option remember;
      `if`(x<0 or y b(n$2, true):
seq(a(n), n=0..40);

Mathematica

b[x_, y_, u_] := b[x, y, u] = If[x<0 || yJean-François Alcover, Mar 24 2017, translated from Maple *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 2.1792514215908330337..., c = 0.4751731999905254789... . - Vaclav Kotesovec, Sep 02 2014