A212390 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 10).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 13, 79, 365, 1366, 4369, 12377, 31825, 75583, 167961, 352718, 705466, 1352585, 2501205, 4495351, 7956391, 14221936, 26802361, 56058016, 133316626, 350785307, 967683665, 2677259721, 7246005881, 18977267621, 47931495649

Offset: 0

Author

Alois P. Heinz, May 12 2012

Keywords

• nonn

Comments

Lengths of descents are unrestricted.

Examples

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(11) = 2: UDUDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUUDDDDDDDDDDD.
a(12) = 13: UDUDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUUDDDDDDDDDDD, UUUUUUUUUUUDDDDDDDDDDDUD, UUUUUUUUUUUDDDDDDDDDDUDD, UUUUUUUUUUUDDDDDDDDDUDDD, UUUUUUUUUUUDDDDDDDDUDDDD, UUUUUUUUUUUDDDDDDDUDDDDD, UUUUUUUUUUUDDDDDDUDDDDDD, UUUUUUUUUUUDDDDDUDDDDDDD, UUUUUUUUUUUDDDDUDDDDDDDD, UUUUUUUUUUUDDDUDDDDDDDDD, UUUUUUUUUUUDDUDDDDDDDDDD, UUUUUUUUUUUDUDDDDDDDDDDD.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Asymptotic of subsequences of A212382

Crossrefs

Column k=10 of A212382.

Programs

Maple

b:= proc(x, y, u) option remember;
      `if`(x<0 or y b(n$2, true):
seq(a(n), n=0..40);
# second Maple program:
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^10), A), x, n+1), x, n):
seq(a(n), n=0..40);

Formula

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^10).

a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where p = 10 and r = 0.421937635689419083..., s = 1.885352542104400040... are roots of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014