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A123746
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Numerators of partial sums of a series for 1/sqrt(2).
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3
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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a(3)=9 because r(n)=1-1/2+3/8-5/16 = 9/16 = a(3)/A046161(3).
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MAPLE
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a := n -> numer(add(binomial(-1/2, j), j=0..n));
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MATHEMATICA
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Table[Numerator[Sum[Binomial[2*k, k]/(-4)^k, {k, 0, n}]], {n, 0, 30}] (* G. C. Greubel, Mar 28 2018 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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