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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 10 2012
STATUS
approved