A124397 -id:A124397

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A124398

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

+10
2

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

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OFFSET

0,2

COMMENTS

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

Numerators are given by A124397.

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

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

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} (-1)^k * binomial(2*k,k)/5^k, in lowest terms.

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.

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

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

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

(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

(GAP) List([0..20], n-> DenominatorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # G. C. Greubel, Dec 25 2019

CROSSREFS

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

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 10 2006

STATUS

approved

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