A124396 Denominators of partial sums of a series for 3/sqrt(5) = (3/5)*sqrt(5).

1, 9, 27, 729, 6561, 6561, 177147, 1594323, 4782969, 387420489, 3486784401, 10460353203, 282429536481, 2541865828329, 2541865828329, 22876792454961, 205891132094649, 617673396283947, 50031545098999707, 450283905890997363

Offset: 0

Author

Wolfdieter Lang, Nov 10 2006

Keywords

• nonn
• frac
• easy

Comments

Denominators of sums over central binomial coefficients scaled by powers of 9.

Numerators are given by A123749.

For the rationals r(n) see the W. Lang link under A123749.

This is not 3/5 times the rational sequence A123747/A123748 which converges to sqrt(5).

Examples

			a(3) = 729 because r(3) = 1 + 2/9 + 2/27 + 20/729 = 965/729 = A123749(3)/a(3).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A123749 (numerators).

Cf. A123747/A123748 partial sums for a series for sqrt(5).

Programs

GAP

List([0..20], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k, k)/9^k)) ); # G. C. Greubel, Dec 25 2019

Magma

[Denominator(&+[(k+1)*Catalan(k)/9^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019

Maple

seq(denom(add(binomial(2*k, k)/9^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019

Mathematica

Table[Denominator[Sum[(k+1)*CatalanNumber[k]/9^k, {k,0,n}]], {n,0,20}] (* G. C. Greubel, Dec 25 2019 *)

PARI

a(n) = denominator(sum(k=0, n, binomial(2*k,k)/9^k)); \\ Michel Marcus, Aug 12 2019

Sage

[denominator(sum((k+1)*catalan_number(k)/9^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019

Formula

a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/9^k in lowest terms.

r(n) = Sum_{k=0..n} ((2*k-1)!!/(2*k)!!)*(4/9)^k, n>=0, with the double factorials A001147 and A000165.