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A121839
Decimal expansion of Sum_{k>=1} 1/C(k), where C(k) is a Catalan Number (A000108).
14
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
1,2
LINKS
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
KEYWORD
cons,nonn
AUTHOR
Alexander Adamchuk, Aug 28 2006
STATUS
approved