A145375 Numerators of partial sums of the alternating series of inverse central binomial coefficients.

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

Table of n, a(n) for n=1..24.

C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. See Eq. 13, p. 39.

W. 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=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

A145557/A145558.

Sequence in context: A162587 A033216 A139837 * A086547 A054291 A261086

Adjacent sequences: A145372 A145373 A145374 * A145376 A145377 A145378

Keyword

nonn,frac,easy

Author

Wolfdieter Lang, Oct 17 2008, Nov 17 2008, Nov 25 2008

Status

approved