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

1,2

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