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A121839 Decimal expansion of sum(k>=1, 1/C(k)), where C(k) is a Catalan Number (A000108). 12
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

Table of n, a(n) for n=1..102.

Alexander Adamchuk's post, August 29, 2006 Mathematics in Russian

Eric Weisstein's World of Mathematics, Catalan Number.

FORMULA

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-Fran├žois Alcover, Jun 06 2016

EXAMPLE

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

= 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

CROSSREFS

Cf. A000108, A002390.

Sequence in context: A190281 A107950 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 August 22 12:29 EDT 2017. Contains 290946 sequences.