A121012 Numerators of partial alternating sums of Catalan numbers scaled by powers of 1/(11^2) = 1/121.

1, 120, 14522, 1757157, 212616011, 25726537289, 282991910191, 34242021133072, 4143284557101842, 501337431409322440, 667280121205808184436, 80740894665902790257970, 9769648254574237621422382

Offset

0,2

Comments

Denominators are given under A121013.

This is the second member (p=2) 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.

Links

Table of n, a(n) for n=0..12.

W. Lang: Rationals r(n), limit.

Formula

a(n)=numerator(r(n)) with r(n) := rIV(p=2,n) = sum(((-1)^k)*C(k)/L(2*2+1)^(2*k), k=0..n), with L(5)=11 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.

Example

Rationals r(n): [1, 120/121, 14522/14641, 1757157/1771561,
212616011/214358881, 25726537289/25937424601,...].

Maple

The limit lim_{n->infinity} (r(n) := rIV(2; n)) = 11*(-8 + 5*phi) = 11/phi^5 = 0.9918693812443 (maple10, 10 digits).

Crossrefs

The first member is A120794/A120785. The third member is A121498/A121499.

Sequence in context: A224390 A223992 A224023 * A250511 A151985 A229605

Adjacent sequences: A121009 A121010 A121011 * A121013 A121014 A121015

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Aug 16 2006

Status

approved