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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)*log(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 formulas 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 formulas...Footnote to Apery's proof of the irrationality of zeta(3), SÃ©minaire Delange-Pisot-Poitou. ThÃ©orie 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 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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