A121839 - OEIS

%I #66 Sep 08 2022 08:45:27

%S 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,

%T 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,

%U 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

%N Decimal expansion of Sum_{k>=1} 1/C(k), where C(k) is a Catalan Number (A000108).

%H G. C. Greubel, <a href="/A121839/b121839.txt">Table of n, a(n) for n = 1..10000</a>

%H Alexander Adamchuk's post, <a href="http://ru-math.livejournal.com/399814.html">Mathematics in Russian</a>, August 29 2006.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>.

%F Reciprocal Catalan Constant C = 1 + 4*sqrt(3)*Pi/27.

%F 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

%F 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

%F From _Vaclav Kotesovec_, May 31 2015: (Start)

%F Equals 1 + Integral_{x=0..1} Product_{k>=1} (1-x^(9*k))^3 dx.

%F Equals 1 + Sum_{n>=0} (-1)^n * (2*n+1) / (9*n*(n+1)/2 + 1).

%F (End)

%F Equals 1 + Integral_{0..inf} x^3 BesselI_0(x) BesselK_0(x)^2 dx. - _Jean-François Alcover_, Jun 06 2016

%F From _Amiram Eldar_, Jul 05 2020: (Start)

%F Equals 1 + gamma(4/3)*gamma(5/3).

%F Equals 1 + Integral_{x=0..oo} dx/(1 + x^3)^2. (End)

%e 1.806133050770763489152923670063180325459584999152...

%p evalf(1 + Sum((-1)^n*(2*n+1)/(9*n*(n+1)/2+1), n=0..infinity), 120); # _Vaclav Kotesovec_, May 31 2015

%t RealDigits[N[Sum[n!(n + 1)!/(2n)!, {n, 1, Infinity}], 150]]

%t RealDigits[N[1+4*Sqrt[3]*Pi/27,100]][[1]]

%o (PARI) default(realprecision,100); 1 + 4*sqrt(3)*Pi/27

%o (Magma) SetDefaultRealField(RealField(100)); R:=RealField(); 1 + 4*Sqrt(3)*Pi(R)/27; // _G. C. Greubel_, Nov 04 2018

%Y Cf. A000108, A002390, A268813 (essentially the same).

%K cons,nonn

%O 1,2

%A _Alexander Adamchuk_, Aug 28 2006