A120786 Numerators of partial sums of Catalan numbers scaled by powers of 1/20.

1, 21, 211, 1689, 84457, 1689161, 16891643, 1351331869, 2702663881, 270266390531, 2702663909509, 108106556409753, 1081065564149533, 4324262256635277, 43242622566419631, 6918819610629079929

Offset: 0

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac

Comments

Denominators are given under A120787.

From the expansion of 2*sqrt(5)/5 = sqrt(1-1/5) = 1-(1/10)*Sum_{k>=0} C(k)/20^k one has r := lim_{n->oo} r(n) = 2*(5 - 2*sqrt(5)) = 2*(7 - 4*phi) = 1.055728090..., where phi := (1+sqrt(5))/2 (golden section) and the partial sums r(n) are defined below.

This is the second member (p=1) in the second p-family of partial sums of the normalized scaled Catalan series CsnII(p) := Sum_{k>=0} C(k)/((5^k)*F(2*p+1)^(2*k)) with limit F(2*p+1)*(L(2*p+2) - L(2*p+1)*phi), 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 second p-family are rII(p;n) := Sum_{k=0..n} C(k)/((5^k)*F(2*p+1)^(2*k)), n>=0, for p=0,1,...

For more details about this p-family and the other three ones see the W. Lang link under A120996.

Examples

			Rationals r(n): [1, 21/20, 211/200, 1689/1600, 84457/80000, 1689161/1600000, 16891643/16000000, 1351331869/1280000000,...].
		

Links

• Wolfdieter Lang, Rationals r(n) and limit.

Formula

a(n) = numerator(r(n)), with the rationals r(n) := Sum_{k=0..n} C(k)/20^k with C(k) := A000108(k) (Catalan numbers). Rationals r(n) are taken in lowest terms.