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A145557 Numerators of partial sums of a certain alternating series of inverse central binomial coefficients. 6
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; text; internal format)
OFFSET

1,2

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.

LINKS

Robert Israel, Table of n, a(n) for n = 1..118

C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45. See Eq. 12, p. 39.

M. L. Glasser, A Generalized Apery Series, Journal of Integer Sequences, Vol. 15 (2012), #12.4.3.

W. Lang, Rationals and more.

Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, Integers: electronic journal of combinatorial number theory, 6 (2006) #A27, 1-18.

A. J. van der Poorten, Some wonderful formulae...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.

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).

EXAMPLE

Rationals r(n) (in lowest terms): [1/2, 5/12, 13/30, 361/840, 31/72, 1193/2772, 31021/72072,...].

MAPLE

R:= 0;

for n from 1 to 100 do

   R:= R + (-1)^(n+1)/(n*binomial(2*n, n));

   a[n]:=numer(R);

od:

seq(a[i], i=1..100); # Robert Israel, Jun 16 2014

PROG

(PARI) vector(50, n, numerator(sum(k=1, n, (-1)^(k+1)/(k*binomial(2*k, k))))) \\ Michel Marcus, Oct 13 2014

CROSSREFS

A145375/A145556.

Sequence in context: A226664 A067135 A122900 * A012033 A007540 A157250

Adjacent sequences:  A145554 A145555 A145556 * A145558 A145559 A145560

KEYWORD

nonn,frac,easy

AUTHOR

Wolfdieter Lang, Oct 17 2008

STATUS

approved

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Last modified November 25 18:10 EST 2014. Contains 250000 sequences.