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

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