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A145559
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Numerators of partial sums of a certain alternating series of inverse central binomial coefficients.
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2
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1, 11, 167, 4667, 7781, 770269, 70095379, 280380781, 14299427671, 271689093997, 229890777659, 68737342138891, 7770308251333, 893585448657907, 43189963354470841, 5355555455879234209, 10116049194470941417, 819399984751544533657, 576038189280433285982311
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| See A145560 for the denominators divided by 2.
The limit of the rational partial sums r(n), defined below, for n->infinity is 2*ln(phi)^2, with phi:=(1+sqrt(5))/2 (golden section). This limit is approximately 0.4631296414.
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REFERENCES
| C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. Eq.11, p.39.
A. J. van der Poorten, Some wonderful formulae...Footnote to Apery's proof of the irrationality of zeta(3), S\'eminaire Delange-Pisot-Poitou. Th\'eorie des nombres, tome 20, no 2 (1978-1979), exp, no 29, p.1-7. p. 29-02 Available via http://www.numdam.org/numdam-bin/qrech
R. Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers: electronic journal of combinatorial number theory, 6 (2006) #A27, 1-18.
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LINKS
| W. Lang, Rationals and more.
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FORMULA
| a(n)=numerator(r(n)) with the rationals (in lowest terms) r(n):=sum(((-1)^(k+1))/(binomial(2*k,k)*k^2),k=1..n).
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EXAMPLE
| Rationals r(n) (in lowest terms): [1/2, 11/24, 167/360, 4667/10080, 7781/16800, 770269/1663200, 70095379/151351200,...].
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MATHEMATICA
| Numerator[Table[Sum[((-1)^(k+1))/(Binomial[2k, k]k^2), {k, n}], {n, 25}]] (* From Harvey P. Dale, Aug 10 2011 *)
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CROSSREFS
| A145557/A145558.
Sequence in context: A142513 A075141 A088293 * A159968 A059091 A157944
Adjacent sequences: A145556 A145557 A145558 * A145560 A145561 A145562
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KEYWORD
| nonn,frac,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 17 2008
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