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

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