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A210737 Number of Dyck n-paths all of whose ascents have prime lengths. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A290679 A290423 A272614 * A140539 A056188 A020696

Adjacent sequences:  A210734 A210735 A210736 * A210738 A210739 A210740

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 10 2012

STATUS

approved

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Last modified February 22 17:35 EST 2019. Contains 320400 sequences. (Running on oeis4.)