A121000 Numerators of partial sums of Catalan numbers scaled by powers of 1/18^2 = 1/324.

1, 325, 52651, 34117853, 5527092193, 596925956851, 96702005009873, 125325798492795551, 60908338067498638501, 19734301533869558876755, 3196956848486868538038509, 2071628037819490812648983225

Offset: 0

Author

Wolfdieter Lang, Aug 16 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators are given under A121001.

This is the fourth member (p=3) of the first p-family of partial sums of normalized scaled Catalan series CsnI(p):=sum(C(k)/L(2*p)^(2*k),k=0..infinity) with limit L(2*p)*(F(2*p+1) - F(2*p)*phi) = L(2*p)/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 first p-family are rI(p;n):=sum(C(k)/L(2*p)^(2*k),k=0..n), n>=0, for p=0,1,...

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) = 18*(13 - 8* phi) = 18/phi^6 = 1.003105620014 (maple10, 15 digits).

Examples

			Rationals r(n): [1, 325/324, 52651/52488, 34117853/34012224,
5527092193/5509980288, 596925956851/595077871104, ...].
		

Links

• W. Lang: Rationals r(n), limit.

Formula

a(n)=numerator(r(n)) with r(n) := rI(p=3,n) = sum(C(k)/L(6)^(2*k),k=0..n), with Lucas L(6)=18 and C(k):=A000108(k) (Catalan). The rationals r(n) are given in lowest terms.