A123746 Numerators of partial sums of a series for 1/sqrt(2).

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

Offset: 0

Author

Wolfdieter Lang, Nov 10 2006

Keywords

• nonn
• frac
• easy

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

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Wolfdieter Lang, Rationals and more.
• Michael Milgram, An Extension of Glasser's Master Theorem and a Collection of Improper Integrals Many of Which Involve Riemann's Zeta Function, ResearchGate, 2024. See p. 20.

Crossrefs

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

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

Programs

GAP

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

Magma

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

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 *)

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

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

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