A120782 Numerators of partial sums of Catalan numbers scaled by powers of 1/12.

1, 13, 79, 1901, 11413, 45659, 273965, 13150463, 236709049, 2840511019, 17043070313, 409033716905, 2454202353433, 29450428426921, 58900856965277, 1884827423966069, 11308964545760729, 407122723668993709

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators are given under A120783.

From the expansion of sqrt(6)/3 = sqrt(1-1/3) = 1-(1/6)*Sum_{k=0..infinity} C(k)/12^k one has r:=limit(r(n),n to infinity)= 2*(3 - sqrt(6)) = 1.101020514..., with the partial sums r(n) defined below.

Examples

			Rationals r(n): [1, 13/12, 79/72, 1901/1728, 11413/10368, 45659/41472, 273965/248832, 13150463/11943936,...].
		

Links

• Wolfdieter Lang, Rationals r(n) and limit

Crossrefs

Cf. A000108, A120783.

Programs

PARI

C(n) = binomial(2*n, n)/(n+1); \\ A000108
a(n) = numerator(sum(k=0, n, C(k)/12^k)); \\ Michel Marcus, Mar 02 2016

Formula

a(n) = numerator(r(n)), with the rationals r(n) = Sum_{k=0..n} C(k)/12^k with C(k) = A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.