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
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.
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) = 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
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Nov 10 2006
STATUS
approved