login
Denominators of partial sums of a series for sqrt(5)/3.
2

%I #11 Sep 08 2022 08:45:28

%S 1,5,25,25,125,3125,15625,78125,78125,390625,9765625,48828125,

%T 244140625,244140625,48828125,6103515625,30517578125,152587890625,

%U 152587890625,762939453125,19073486328125,95367431640625,476837158203125

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

%C Denominators of alternating sums over central binomial coefficients scaled by powers of 5.

%C Numerators are given by A124397.

%C For the rationals r(n) see the W. Lang link under A124397.

%C r(n) is not 1/3 times the rational sequence A123747/A123748 which converges to sqrt(5).

%H G. C. Greubel, <a href="/A124398/b124398.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} (-1)^k * binomial(2*k,k)/5^k, in lowest terms.

%F r(n) = Sum_{k=0..n} (-1)^k*((2*k-1)!!/((2*k)!!)*(4/5)^k, n>=0, with the double factorials A001147 and A000165.

%e a(3) = 25 because r(3)= 1 - 2/5 + 6/25 - 4/25 = 17/25 = A124397(3)/a(3).

%p seq(denom(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # _G. C. Greubel_, Dec 25 2019

%t Table[Denominator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k,0,n}]], {n,0,20}] (* _G. C. Greubel_, Dec 25 2019 *)

%o (PARI) a(n) = denominator(sum(k=0, n, ((-1)^k)*binomial(2*k,k)/5^k)); \\ _Michel Marcus_, Aug 11 2019

%o (Magma) [Denominator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Dec 25 2019

%o (Sage) [denominator(sum((-1)^k*(k+1)*catalan_number(k)/5^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-> (-1)^k*Binomial(2*k, k)/5^k)) ); # _G. C. Greubel_, Dec 25 2019

%Y Cf. A124397 (numerators), A208899 (sqrt(5)/3).

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 10 2006