A120782 - OEIS

%I #19 Jun 13 2022 07:48:41

%S 1,13,79,1901,11413,45659,273965,13150463,236709049,2840511019,

%T 17043070313,409033716905,2454202353433,29450428426921,58900856965277,

%U 1884827423966069,11308964545760729,407122723668993709

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

%C Denominators are given under A120783.

%C 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.

%H Wolfdieter Lang, <a href="/A120782/a120782.txt">Rationals r(n) and limit</a>

%F 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.

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

%o (PARI) C(n) = binomial(2*n, n)/(n+1); \\ A000108

%o a(n) = numerator(sum(k=0, n, C(k)/12^k)); \\ _Michel Marcus_, Mar 02 2016

%Y Cf. A000108, A120783.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Jul 20 2006