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

0,5

Links

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

Formula

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

Example

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.

Maple

with(numtheory):
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-ithprime(t), y, false), t=1..pi(x)), 0)))
    end:
a:= n-> b(n, n, true):
seq(a(n), n=0..40);

Mathematica

b[x_, y_, u_] := b[x, y, u] = If[x<0 || y<x, 0, If[x == 0 && y == 0, 1, b[x, y-1, True] + If[u, Sum [b[x-Prime[t], y, False], {t, 1, PrimePi[x]}], 0]]]; a[n_] := b[n, n, True]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Prog

(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

Crossrefs

Cf. A210735.

Sequence in context: A290423 A368538 A272614 * A140539 A056188 A020696

Adjacent sequences: A210734 A210735 A210736 * A210738 A210739 A210740

Keyword

nonn

Author

Alois P. Heinz, May 10 2012

Status

approved