%I #13 Sep 15 2024 02:47:21

%S 1,15,121,3867,30943,495067,3960569,253475987,2027808611,32444935345,

%T 259559486959,8305903553295,66447228478363,1063155655468083,

%U 8505245244078969,1088671391232413187

%N Numerators of partial sums of Catalan numbers scaled by powers of -1/16.

%C From the expansion of sqrt(1+1/4) = 1+(1/8)*Sum_{k>=0} C(k)/(-16)^k one has, with the partial sums r(n) are defined below, r := lim_{n->oo} r(n) = 4*(sqrt(5)-2) = 4*(2*phi-3) = 0.944271909...

%C Denominators coincide with the listed numbers of A120785 but may differ for higher n values.

%C This is the first member (p=1) of the fourth family of scaled Catalan sums with limits in Q(sqrt(5)). See the W. Lang link under A120996.

%H Wolfdieter Lang, <a href="/A120794/a120794.txt">Rationals r(n) and limit.</a>

%F a(n)=numerator(r(n)), with the rationals r(n):=Sum_{k=0..n} ((-1)^k)*C(k)/16^k with C(k):=A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.

%e Rationals r(n): [1, 15/16, 121/128, 3867/4096, 30943/32768, 495067/524288, 3960569/4194304,...].

%Y The second member (p=2) of this p-family is A121012/A121013.

%K nonn,easy,frac

%O 0,2

%A _Wolfdieter Lang_, Jul 20 2006