A121008 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(5*3^2) = 1/45.

1, 44, 1982, 17837, 4013339, 60200071, 2709003239, 121905145612, 658287786362, 740573759652388, 33325819184374256, 1499661863296782734, 67484783848355431042, 607363054635198730798, 3036815273175993713422

Offset: 0

Author

Wolfdieter Lang, Aug 16 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators are given under A121009.

This is the second member (p=2) 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.

Examples

			Rationals r(n): [1, 44/45, 1982/2025, 17837/18225, 4013339/4100625,
60200071/61509375, 2709003239/2767921875,...].
		

Links

• W. Lang: Rationals r(n), limit.

Crossrefs

The first member (p=1) is A121006/A121007.

Programs

Maple

The limit lim_{n->infinity}(r(n) := rIII(2;n)) = 3*(-11 + 7*phi) = 3*sqrt(5)/phi^4 = 0.9787137637479 (maple10, 15 digits).

Formula

a(n)=numerator(r(n)) with r(n) := rIII(p=2,n) = sum(((-1)^k)*C(k)/((5^k)*F(2*2)^(2*k)),k=0..n), with F(4)=3 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.