

A120794


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


4



1, 15, 121, 3867, 30943, 495067, 3960569, 253475987, 2027808611, 32444935345, 259559486959, 8305903553295, 66447228478363, 1063155655468083, 8505245244078969, 1088671391232413187
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OFFSET

0,2


COMMENTS

From the expansion of sqrt(1+1/4) = 1+(1/8)*sum(C(k)/(16)^k,k=0..infinity) one has, with the partial sums r(n) are defined below, r:=limit(r(n),n to infinity)= 4*(sqrt(5)2) = 4*(2*phi3)) = 0.944271909...
Denominators coincide with the listed numbers of A120785 but may differ for higher n values.
This is the first member (p=1) of the fourth famliy of scaled Catalan sums with limits in Q(sqrt(5)). See the W. Lang link under A120996.


LINKS



FORMULA

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


EXAMPLE

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


CROSSREFS



KEYWORD

nonn,easy,frac


AUTHOR



STATUS

approved



