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

%I #17 Jan 26 2025 09:09:59

%S 1,9,27,729,6561,6561,177147,1594323,4782969,387420489,3486784401,

%T 10460353203,282429536481,2541865828329,2541865828329,22876792454961,

%U 205891132094649,617673396283947,50031545098999707,450283905890997363

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

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

%C Numerators are given by A123749.

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

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

%H G. C. Greubel, <a href="/A124396/b124396.txt">Table of n, a(n) for n = 0..1000</a>

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

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

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

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

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

%o (PARI) a(n) = denominator(sum(k=0, n, binomial(2*k,k)/9^k)); \\ _Michel Marcus_, Aug 12 2019

%o (Magma) [Denominator(&+[(k+1)*Catalan(k)/9^k: k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Dec 25 2019

%o (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

%o (GAP) List([0..20], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k, k)/9^k)) ); # _G. C. Greubel_, Dec 25 2019

%Y Cf. A123749 (numerators).

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

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 10 2006