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 A121839 Decimal expansion of Sum_{k>=1} 1/C(k), where C(k) is a Catalan Number (A000108). 13
 1, 8, 0, 6, 1, 3, 3, 0, 5, 0, 7, 7, 0, 7, 6, 3, 4, 8, 9, 1, 5, 2, 9, 2, 3, 6, 7, 0, 0, 6, 3, 1, 8, 0, 3, 2, 5, 4, 5, 9, 5, 8, 4, 9, 9, 9, 1, 5, 2, 3, 2, 9, 1, 4, 4, 6, 9, 7, 7, 2, 6, 6, 3, 7, 9, 5, 0, 2, 7, 6, 9, 6, 9, 3, 8, 9, 4, 9, 0, 6, 1, 4, 9, 7, 0, 7, 2, 2, 2, 1, 6, 9, 8, 3, 1, 3, 7, 8, 5, 2, 8, 2 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS G. C. Greubel, Table of n, a(n) for n = 1..10000 Alexander Adamchuk's post, Mathematics in Russian, August 29 2006. Eric Weisstein's World of Mathematics, Catalan Number. FORMULA Reciprocal Catalan Constant C = 1 + 4*sqrt(3)*Pi/27. This number is f(1) where f(x) = -1 + 2*(sqrt(4-x)*(8+x) + 12 * sqrt(x) * arctan(sqrt(x)/sqrt(4-x))) / sqrt((4-x)^5). This form corresponds to a generating function of the reciprocal Catalan numbers in the sense of Sprugnoli. - Juan M. Marquez, Mar 05 2009 Equals -1 + hypergeom([1,2],[1/2],1/4); note hypergeom([1,2],[1/2],x/4) = 1/1 + 1/1*x + 1/2*x^2 + 1/5*x^3 + 1/14*x^4 + 1/42*x^5 + ... is the g.f. for the inverse Catalan numbers (including C(0)). - Joerg Arndt, Apr 06 2013 From Vaclav Kotesovec, May 31 2015: (Start) Equals 1 + Integral_{x=0..1} Product_{k>=1} (1-x^(9*k))^3 dx. Equals 1 + Sum_{n>=0} (-1)^n * (2*n+1) / (9*n*(n+1)/2 + 1). (End) Equals 1 + Integral_{0..inf} x^3 BesselI_0(x) BesselK_0(x)^2 dx. - Jean-François Alcover, Jun 06 2016 From Amiram Eldar, Jul 05 2020: (Start) Equals 1 + gamma(4/3)*gamma(5/3). Equals 1 + Integral_{x=0..oo} dx/(1 + x^3)^2. (End) EXAMPLE 1.806133050770763489152923670063180325459584999152... MAPLE evalf(1 + Sum((-1)^n*(2*n+1)/(9*n*(n+1)/2+1), n=0..infinity), 120); # Vaclav Kotesovec, May 31 2015 MATHEMATICA RealDigits[N[Sum[n!(n + 1)!/(2n)!, {n, 1, Infinity}], 150]] RealDigits[N[1+4*Sqrt[3]*Pi/27, 100]][[1]] PROG (PARI) default(realprecision, 100); 1 + 4*sqrt(3)*Pi/27 (MAGMA) SetDefaultRealField(RealField(100)); R:=RealField(); 1 + 4*Sqrt(3)*Pi(R)/27; // G. C. Greubel, Nov 04 2018 CROSSREFS Cf. A000108, A002390, A268813 (essentially the same). Sequence in context: A107950 A336798 A273634 * A010517 A021851 A021996 Adjacent sequences:  A121836 A121837 A121838 * A121840 A121841 A121842 KEYWORD cons,nonn AUTHOR Alexander Adamchuk, Aug 28 2006 STATUS approved

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Last modified May 18 23:31 EDT 2022. Contains 353826 sequences. (Running on oeis4.)