A210453 - OEIS

A210453

Decimal expansion of sum_{n>=1} 1/(n*binomial(3*n,n)).

3, 7, 1, 2, 1, 6, 9, 7, 5, 2, 6, 0, 2, 4, 7, 0, 3, 4, 4, 7, 4, 7, 7, 1, 6, 6, 6, 0, 7, 5, 3, 5, 8, 8, 0, 5, 5, 8, 7, 6, 2, 9, 4, 6, 9, 0, 5, 1, 9, 7, 2, 2, 2, 1, 3, 6, 4, 7, 7, 8, 9, 3, 9, 5, 7, 3, 4, 0, 0, 0, 8, 3, 5, 3, 5, 5, 9, 8, 4, 9, 6, 9, 1, 3, 1, 4, 3, 2, 7, 5, 4, 1, 7, 7, 6, 5, 0, 5, 0, 9, 9, 2, 3, 2, 3, 9, 6, 1, 7, 5, 6, 9, 0, 7, 7, 3, 5, 3, 5, 2, 7, 3, 1, 6, 8, 6

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OFFSET

0,1

COMMENTS

Equals the integral over x^2/(1-x^2+x^3) dx between x=0 and x=1.

REFERENCES

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 60.

LINKS

Table of n, a(n) for n=0..125.

Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991) 53-55.

FORMULA

Equals sum_(R) R*log(1-1/R)/(3*R-2) where R is summed over the set of the three constants -A075778, A210462-i*A210463 and A210462-i*A210463, i=sqrt(-1), that is, over the set of the three roots of x^3-x^2+1.

EXAMPLE

0.371216975260247034474771... = sum_{n>=1} 1/(n*A005809(n)).

MAPLE

A075778neg := proc()

1/3-root[3](25/2-3*sqrt(69)/2)/3 -root[3](25/2+3*sqrt(69)/2)/3;

end proc:

A210462 := proc()

local a075778 ;

a075778 := A075778neg() ;

(1+1/a075778/(a075778-1))/2 ;

end proc:

A210463 := proc()

local a075778, a210462 ;

a075778 := A075778neg() ;

a210462 := A210462() ;

-1/a075778-a210462^2 ;

sqrt(%) ;

end proc:

A210453 := proc()

local v, x;

v := 0.0 ;

for x in [ A075778neg(), A210462()+I*A210463(), A210462()-I*A210463() ] do

v := v+ x*log(1-1/x)/(3*x-2) ;

end do:

evalf(v) ;

end proc:

A210453() ;

MATHEMATICA

RealDigits[ HypergeometricPFQ[{1, 1, 3/2}, {4/3, 5/3}, 4/27]/3, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

CROSSREFS

Cf. A073010

Sequence in context: A127929 A228147 A200922 * A003118 A096409 A356704

Adjacent sequences: A210450 A210451 A210452 * A210454 A210455 A210456

KEYWORD

cons,nonn

AUTHOR

R. J. Mathar, Jan 21 2013

STATUS

approved