A121498 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(29^2) = 1/841.

1, 840, 706442, 594117717, 499653000011, 420208173009209, 353395073500744901, 297205256814126461312, 249949620980680353964822, 210207631244752177684410440, 176784617876836581432589196836

Offset: 0

Author

Wolfdieter Lang, Aug 16 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators are given under A121499.

This is the third member (p=3) of the fourth (normalized) p-family of partial sums of normalized scaled Catalan series CsnIV(p):=sum(((-1)^k)*C(k)/L(2*p+1)^(2*k),k=0..infinity) with limit L(2*p+1)*(-F(2*p+2) + F(2*p+1)*phi) = L(2*p+1)/phi^(2*p+1), 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 fourth p-family are rIV(p;n):=sum(((-1)^k)*C(k)/L(2*p+1)^(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, 840/841, 706442/707281, 594117717/594823321,
499653000011/500246412961, 420208173009209/420707233300201,...].
		

Links

• W. Lang: Rationals r(n), limit.

Crossrefs

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

Programs

Maple

The limit lim_{n->infinity}(r(n) := rIV(2;n)) = 29*(-21 + 13*phi) = 29/phi^7 = 0.998813758709 (maple10, 10 digits).

Formula

a(n)=numerator(r(n)) with r(n) := rIV(p=3,n) = sum(((-1)^k)*C(k)/L(2*3+1)^(2*k),k=0..n), with L(7)=29 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.