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A123746 Numerators of partial sums of a series for 1/sqrt(2). 3
1, 1, 7, 9, 107, 151, 835, 1241, 26291, 40427, 207897, 327615, 3296959, 5293843, 26189947, 42685049, 1666461763, 2749521971, 13266871709, 22115585443, 211386315749, 355490397193, 1684973959237, 2855358497999, 53747636888759 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Denominators are given by A046161(n),n>=0.

The alternating sum over central binomial coefficients scaled by powers of 4, r(n) = Sum_{k=0..n} (-1)^k*binomial(2*k,k)/4^k, has the limit s = lim_{n->infinity} r(n) = 1/sqrt(2). From the expansion of 1/sqrt(1-x) for |x|<1 which extends to x=-1 due to Abel's limit theorem and the convergence of the series s. See the W. Lang link.

(2^n)*n!*r(n) = A003148(n). - Wolfdieter Lang, Oct 06 2008

LINKS

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

W. Lang, Rationals and more.

FORMULA

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k* binomial(2*k,k)/4^k, n>=0.

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

r(n) = 1/sqrt(2) - binomial(-1/2, 1 + n)*hypergeom([1, 3/2 + n], [2 + n], -1). - Peter Luschny, Sep 26 2019

EXAMPLE

a(3)=9 because r(n)=1-1/2+3/8-5/16 = 9/16 = a(3)/A046161(3).

MAPLE

A123746:=n-> numer(add(binomial(2*k, k)/(-4)^k, k=0..n)); seq(A123746(n), n=0..30); # G. C. Greubel, Aug 10 2019

a := n -> numer(add(binomial(-1/2, j), j=0..n));

seq(a(n), n=0..24); # Peter Luschny, Sep 26 2019

MATHEMATICA

Table[Numerator[Sum[Binomial[2*k, k]/(-4)^k, {k, 0, n}]], {n, 0, 30}] (* G. C. Greubel, Mar 28 2018 *)

PROG

(PARI) {r(n) = sum(k=0, n, (-1/4)^k*binomial(2*k, k))};

vector(30, n, n--; numerator(r(n)) ) \\ G. C. Greubel, Mar 28 2018

(MAGMA) [Numerator( (&+[Binomial(2*k, k)/(-4)^k: k in [0..n]])): n in [0..30]]; // G. C. Greubel, Aug 10 2019

(Sage) [numerator( sum(binomial(2*k, k)/(-4)^k for k in (0..n)) ) for n in (0..30)] # G. C. Greubel, Aug 10 2019

(GAP) List([0..30], n-> NumeratorRat(Sum([0..n], k-> Binomial(2*k, k)/(-4)^k )) ); # G. C. Greubel, Aug 10 2019

CROSSREFS

Cf. A120088/(2*A120777) partial sums for a series of sqrt(2).

Equals A003148 divided by A049606. - Johannes W. Meijer, Nov 23 2009

Sequence in context: A317401 A137058 A116237 * A152551 A012252 A262538

Adjacent sequences:  A123743 A123744 A123745 * A123747 A123748 A123749

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Nov 10 2006

STATUS

approved

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Last modified July 14 21:41 EDT 2020. Contains 335737 sequences. (Running on oeis4.)