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%I #17 Apr 28 2018 18:17:05
%S 1,0,1,1,2,6,8,29,50,141,327,771,2047,4746,12644,30941,79886,204885,
%T 522242,1365056,3505825,9185742,23907116,62636476,164624803,432540010,
%U 1142827935,3017208675,7996379870,21211540268,56369770281,150086840133,400009010758
%N Number of Dyck n-paths all of whose ascents have prime lengths.
%H Alois P. Heinz, <a href="/A210737/b210737.txt">Table of n, a(n) for n = 0..700</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 2.7925684676903082567..., c = 0.4016264581712556... . - _Vaclav Kotesovec_, Sep 02 2014
%e a(0) = 1: the empty path.
%e a(1) = 0.
%e a(2) = 1: UUDD.
%e a(3) = 1: UUUDDD.
%e a(4) = 2: UUDDUUDD, UUDUUDDD.
%e a(5) = 6: UUDDUUUDDD, UUDUUUDDDD, UUUDDDUUDD, UUUDDUUDDD, UUUDUUDDDD, UUUUUDDDDD.
%e a(6) = 8: UUDDUUDDUUDD, UUDDUUDUUDDD, UUDUUDDDUUDD, UUDUUDDUUDDD, UUDUUDUUDDDD, UUUDDDUUUDDD, UUUDDUUUDDDD, UUUDUUUDDDDD.
%p with(numtheory):
%p b:= proc(x, y, u) option remember;
%p `if`(x<0 or y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p `if`(u, add(b(x-ithprime(t), y, false), t=1..pi(x)), 0)))
%p end:
%p a:= n-> b(n, n, true):
%p seq(a(n), n=0..40);
%t 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_ *)
%o (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
%Y Cf. A210735.
%K nonn
%O 0,5
%A _Alois P. Heinz_, May 10 2012
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