OFFSET
0,2
REFERENCES
George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), Chapter 13 A Master Formula, p. 250.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Beta Function
Eric W. Weisstein, Euler's Series Transformation.
FORMULA
Equals Integral_{x = 0..1} x^2/(1 + x^2)^3 dx.
Also equals beta(3/2, 1/2)/16, where 'beta' is Euler's beta function.
From Peter Bala, Oct 27 2019: (Start)
Equals Integral_{x = 0..1} x^4*sqrt(1 - x^2) dx = Integral_{x = 0..1} x^5*sqrt(1 - x^4) dx = Integral_{x = 0..1} x^7*sqrt(1 - x^16) dx.
Equals Integral_{x >= 0} x^4/(1 + x^2)^4 dx. (End)
From Amiram Eldar, Jul 13 2020: (Start)
Equals Integral_{x=0..oo} dx/(x^2 + 4)^2.
Equals Sum_{k>=1} sin(k)^3*cos(k)^3/k. (End)
From Peter Bala, Dec 08 2021: (Start)
Pi/32 = Sum_{n >= 1} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9)).
Applying Euler's series transformation to this alternating sum gives
Pi/32 = Sum_{n >= 1} 2^(n-3)*n*(n+1)/((2*n+3)*binomial(2*n+2, n+1)). (End)
EXAMPLE
0.0981747704246810387019576057274844651311615437304720569054670185096...
MATHEMATICA
Join[{0}, RealDigits[Pi/32, 10, 105] // First]
PROG
(PARI) Pi/32 \\ Charles R Greathouse IV, Sep 28 2022
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Jul 09 2014
STATUS
approved