A123746 - OEIS

%I #41 Mar 26 2026 15:16:17

%S 1,1,7,9,107,151,835,1241,26291,40427,207897,327615,3296959,5293843,

%T 26189947,42685049,1666461763,2749521971,13266871709,22115585443,

%U 211386315749,355490397193,1684973959237,2855358497999,53747636888759

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

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

%C 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.

%C (2^n)*n!*r(n) = A003148(n). - _Wolfdieter Lang_, Oct 06 2008

%H G. C. Greubel, <a href="/A123746/b123746.txt">Table of n, a(n) for n = 0..1000</a>

%H Wolfdieter Lang, <a href="/A123746/a123746.txt">Rationals and more</a>.

%H Michael Milgram, <a href="https://doi.org/10.13140/RG.2.2.23955.04648">An Extension of Glasser's Master Theorem and a Collection of Improper Integrals Many of Which Involve Riemann's Zeta Function</a>, ResearchGate, 2024. See p. 20.

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

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

%F r(n) = 1/sqrt(2) - binomial(-1/2, 1 + n)*hypergeom([1, 3/2 + n], [2 + n], -1). - _Peter Luschny_, Sep 26 2019

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

%p 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

%p # Alternative:

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

%p seq(a(n), n=0..24); # _Peter Luschny_, Sep 26 2019

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

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

%o vector(30, n, n--; numerator(r(n)) ) \\ _G. C. Greubel_, Mar 28 2018

%o (Magma) [Numerator( (&+[Binomial(2*k,k)/(-4)^k: k in [0..n]])): n in [0..30]]; // _G. C. Greubel_, Aug 10 2019

%o (SageMath) [numerator( sum(binomial(2*k,k)/(-4)^k for k in (0..n)) ) for n in (0..30)] # _G. C. Greubel_, Aug 10 2019

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

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

%Y Equals A003148 divided by A049606. - _Johannes W. Meijer_, Nov 23 2009

%K nonn,frac,easy

%O 0,3

%A _Wolfdieter Lang_, Nov 10 2006