A121839 Decimal expansion of Sum_{k>=1} 1/C(k), where C(k) is a Catalan Number (A000108).
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
Offset: 1
Examples
1.806133050770763489152923670063180325459584999152...
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.
Programs
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Magma
SetDefaultRealField(RealField(100)); R:=RealField(); 1 + 4*Sqrt(3)*Pi(R)/27; // G. C. Greubel, Nov 04 2018
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Maple
evalf(1 + Sum((-1)^n*(2*n+1)/(9*n*(n+1)/2+1), n=0..infinity), 120); # Vaclav Kotesovec, May 31 2015
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Mathematica
RealDigits[N[Sum[n!(n + 1)!/(2n)!, {n, 1, Infinity}], 150]] RealDigits[N[1+4*Sqrt[3]*Pi/27,100]][[1]] -
PARI
default(realprecision,100); 1 + 4*sqrt(3)*Pi/27
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)