A121006 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/5.

1, 4, 22, 21, 539, 2653, 13397, 66556, 66842, 1666188, 8347736, 41679894, 208607482, 208458902, 1042829398, 5212208021, 26068111639, 130314629237, 26066746957, 3257989916987, 16291262409019

Offset: 0

Author

Wolfdieter Lang, Aug 16 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators are given under A121007.

This is the first member (p=1) of the third p-family of partial sums of normalized scaled Catalan series CsnIII(p):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..infinity) with limit F(2*p)*(-L(2*p+1) + L(2*p)*phi)= F(2*p)*sqrt(5)/phi^(2*p), with C(n)=A000108(n) (Catalan), F(n)= A000045(n) (Fibonacci), L(n) = A000032(n) (Lucas) and phi:=(1+sqrt(5))/2 (golden section).

The partial sums of the above mentioned third p-family are rIII(p;n):=sum(((-1)^k)*C(k)/((5^k)*F(2*p)^(2*k)),k=0..n), n>=0, for p=1,...

For more details on this p-family and the other three ones see the W. Lang link under A120996.

The limit lim_{n->infinity} (r(n) := rIII(1;n)) = -4 + 3*phi = sqrt(5)/phi^2 = 0.85410196624968...

Examples

			Rationals r(n): 1, 4/5, 22/25, 21/25, 539/625, 2653/3125,
13397/15625, 66556/78125, 66842/78125, 1666188/1953125, 8347736/9765625,...
		

Links

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

Crossrefs

The second member (p=2) is A121008/A121009.

Formula

a(n)=numerator(r(n)) with r(n) := rIII(p=1,n) = sum(((-1)^k)*C(k)/5^k,k=0..n) and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.