A120794 Numerators of partial sums of Catalan numbers scaled by powers of -1/16.

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

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

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...

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 family of scaled Catalan sums with limits in Q(sqrt(5)). See the W. Lang link under A120996.

Examples

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

Links

• Wolfdieter Lang, Rationals r(n) and limit.

Crossrefs

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

Formula

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.