A123748 Denominators of partial sums of a series for sqrt(5).

1, 5, 25, 5, 125, 3125, 15625, 78125, 78125, 390625, 9765625, 48828125, 244140625, 244140625, 78125, 6103515625, 30517578125, 152587890625, 152587890625, 762939453125, 19073486328125, 95367431640625, 476837158203125, 476837158203125, 2384185791015625

Offset

0,2

Comments

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

Numerators are given by A123747.

For the rationals r(n) see the W. Lang link under A123747.

Links

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

Formula

a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/5^k in lowest terms.

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

Example

a(3) = 5 because r(3) = 1+2/5+6/25+4/25 = 9/5 = A123747(3)/a(3).

Maple

A123748:=n-> denom(sum(binomial(2*k, k)/5^k, k=0..n)); seq(A123748(n), n=0..25); # G. C. Greubel, Aug 10 2019

Mathematica

Table[Denominator[Sum[Binomial[2*k, k]/5^k, {k, 0, n}]], {n, 0, 25}] (* G. C. Greubel, Aug 10 2019 *)

Prog

(PARI) vector(25, n, n--; denominator(sum(k=0, n, binomial(2*k, k)/5^k))) \\ G. C. Greubel, Aug 10 2019
(Magma) [Denominator( (&+[Binomial(2*k, k)/5^k: k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 10 2019
(SageMath) [denominator( sum(binomial(2*k, k)/5^k for k in (0..n)) ) for n in (0..25)] # G. C. Greubel, Aug 10 2019
(GAP) List([0..25], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k, k)/5^k )) ); # G. C. Greubel, Aug 10 2019

Crossrefs

Cf. A123747, A123749.

Sequence in context: A346994 A070387 A331272 * A050108 A070386 A050084

Adjacent sequences: A123745 A123746 A123747 * A123749 A123750 A123751

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Nov 10 2006

Status

approved