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A121839
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Decimal expansion of sum(k>=1, 1/C(k)), where C(k) is a Catalan Number (A000108).
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3
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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
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OFFSET
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1,2
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LINKS
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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.
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FORMULA
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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)
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EXAMPLE
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Reciprocal Catalan Constant C = 1 + 4*sqrt(3)*Pi/27
= 1.806133050770763489152923670063180325459584999152...
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MAPLE
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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
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RealDigits[N[Sum[n!(n + 1)!/(2n)!, {n, 1, Infinity}], 150]]
RealDigits[N[1+4*Sqrt[3]*Pi/27, 100]][[1]]
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PROG
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(PARI) default(realprecision, 100);
1 + 4*sqrt(3)*Pi/27
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CROSSREFS
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Cf. A000108, A002390.
Sequence in context: A132034 A190281 A107950 * A010517 A021851 A021996
Adjacent sequences: A121836 A121837 A121838 * A121840 A121841 A121842
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KEYWORD
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cons,nonn
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AUTHOR
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Alexander Adamchuk, Aug 28 2006
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STATUS
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approved
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