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A145559 Numerators of partial sums of a certain alternating series of inverse central binomial coefficients. 2
1, 11, 167, 4667, 7781, 770269, 70095379, 280380781, 14299427671, 271689093997, 229890777659, 68737342138891, 7770308251333, 893585448657907, 43189963354470841, 5355555455879234209, 10116049194470941417, 819399984751544533657, 576038189280433285982311 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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*log(phi)^2, with phi:=(1+sqrt(5))/2 (golden section). This limit is approximately 0.4631296414.
LINKS
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. See Eq. 11, p. 39.
Wolfdieter Lang, Rationals and more.
A. J. van der Poorten, Some wonderful formulas...Footnote to Apery's proof of the irrationality of zeta(3), Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 20, no. 2 (1978-1979), exp, no. 29, pp. 1-7, pp. 29-02.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers: electronic journal of combinatorial number theory, 6 (2006) #A27, 1-18.
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).
G.f. of r(n): 2*arcsinh(sqrt(x)/2)^2/(1-x). - Robert Israel, Sep 24 2017
EXAMPLE
Rationals r(n) (in lowest terms): [1/2, 11/24, 167/360, 4667/10080, 7781/16800, 770269/1663200, 70095379/151351200,...].
MAPLE
map(numer, ListTools:-PartialSums([seq((-1)^(k+1)/(binomial(2*k, k)*k^2), k=1..30)])); # Robert Israel, Sep 24 2017
MATHEMATICA
Numerator[Table[Sum[((-1)^(k+1))/(Binomial[2k, k]k^2), {k, n}], {n, 25}]] (* Harvey P. Dale, Aug 10 2011 *)
PROG
(PARI) vector(50, n, numerator(sum(k=1, n, ((-1)^(k+1)/(binomial(2*k, k)*k^2))))) \\ Michel Marcus, Oct 13 2014
CROSSREFS
Cf. A145560.
Sequence in context: A142513 A075141 A088293 * A159968 A059091 A255971
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Oct 17 2008
STATUS
approved

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