A120996 Numerators of partial sums of Catalan numbers scaled by powers of 1/9.

1, 10, 92, 833, 7511, 22547, 202967, 1826846, 49326272, 443941310, 3995488586, 35959456060, 323635312552, 2912718555868, 2912718853028, 26214470754457, 235930240718743, 6370116542620991, 57331049042801819

Offset: 0

Author

Wolfdieter Lang, Aug 16 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators are given under A120997.

This is the second member (p=1) 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).

Other members of this rational p-family are: A120778(n)/A120777(n) (p=0), ......

From the expansion of sqrt(1-x) = 1 -(x/2)*sum(C(k)*(x/4)^k,k=0..infinity), for |x|<=1, one has sum(C(k)/q^k,k=0..infinity) = (q - sqrt(q*(q-4)))/2, for |q|>=4.

The CsnI(1) series value is lim_{n->infinity}(r(n)) = 3*(2-phi) = 3/phi^2 = 1.145898034 (maple10, 10 digits).

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 special q values q(n):=2 + L(2*n), n>=0, one finds the above given limits for the p-family members for n=2*p (even), p>=0. The Pell equation x^2 -5*y^2 = +4 with general (nonnegative) solutions (x,y)=(L(2*n),F(2*n)) and well known identities for Fibonacci and Lucas numbers were used to derive the above given p-family result. See the W. Lang link.

Examples

			Rationals r(n): [1, 10/9, 92/81, 833/729, 7511/6561, 22547/19683,
202967/177147, 1826846/1594323,...].
		

Links

• W. Lang: Rationals r(n), limit and four p-families of scaled Catalan series.

Formula

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