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A145375
Numerators of partial sums of the alternating series of inverse central binomial coefficients.
4
1, 1, 23, 31, 47, 1031, 26827, 134107, 455989, 8663665, 4331849, 187279, 4981622687, 747243353, 173360460899, 1074834852769, 233659750871, 926581770421, 198844447947463, 6856705101503, 1630524473145553, 350562761725846217, 97378544923877951, 42247307182355837
OFFSET
1,3
COMMENTS
See A145556 for the denominators.
The limit of the rational partial sums r(n), defined below, for n->infinity is (1 + 4*log(phi)/(2*phi-1))/5, with phi:=(1+sqrt(5))/2 (golden section). This limit is approximately 0.3721635764.
LINKS
C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. See Eq. 13, p. 39.
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=1..n).
EXAMPLE
Rationals r(n) (in lowest terms): [1/2, 1/3, 23/60, 31/84, 47/126, 1031/2772, 26827/72072, ...].
PROG
(PARI) vector(50, n, numerator(sum(k=1, n, (-1)^(k+1)/binomial(2*k, k)))) \\ Michel Marcus, Oct 13 2014
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Wolfdieter Lang, Oct 17 2008, Nov 17 2008, Nov 25 2008
STATUS
approved