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A145557
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Numerators of partial sums of a certain alternating series of inverse central binomial coefficients.
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6
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1, 5, 13, 361, 31, 1193, 31021, 34467, 5273479, 1821745, 220211, 230450795, 2880634987, 1502939987, 5896829249, 12430516053889, 1381168450513, 3271188435379, 2299645470079393, 459929094015491, 819873602375609, 810854992749436603, 311867304903633289
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| See A145558 for the denominators divided by 2.
The limit of the rational partial sums r(n), defined below, for n->infinity is 2*(2*phi-1)*ln(phi)/5, with phi:=(1+sqrt(5))/2 (golden section). This limit is approximately 0.4304089412.
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REFERENCES
| C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. Eq.12, 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))/(k*binomial(2*k,k)),k=1..n).
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EXAMPLE
| Rationals r(n) (in lowest terms): [1/2, 5/12, 13/30, 361/840, 31/72, 1193/2772, 31021/72072,...].
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CROSSREFS
| A145375/A145556.
Sequence in context: A085554 A067135 A122900 * A012033 A007540 A157250
Adjacent sequences: A145554 A145555 A145556 * A145558 A145559 A145560
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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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