A258232 - OEIS

%I #28 Feb 04 2024 03:26:02

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

%T 1,5,0,1,4,2,4,1,3,0,3,9,2,8,8,1,9,9,6,8,3,0,3,6,1,5,8,0,6,6,8,2,8,1,

%U 4,7,3,0,0,8,8,9,0,3,4,3,9,2,9,8,9,0,6,3,4,4,2,4,2,4,1,4,9,9,2,1,7,6,7,1,2,8

%N Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k) dx.

%D József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 424.

%H Jonathan M. Borwein and Peter B. Borwein, <a href="https://doi.org/10.2307/2324993">Strange series and high precision fraud</a>, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640; <a href="https://carmamaths.org/resources/jon/Preprints/Books/MbyE/Second-Ed/Material/strange-series.pdf">alternative link</a>.

%H Martin Klazar, <a href="https://arxiv.org/abs/1808.08449">What is an answer? - remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018.

%H Vaclav Kotesovec, <a href="/A258232/a258232_2.pdf">The integration of q-series</a>.

%F Equals 8*Pi*sqrt(3/23) * sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3) - 1).

%F From _Amiram Eldar_, Feb 04 2024: (Start)

%F Equals 2 * Sum_{k=-oo..oo} (-1)^k/(3*k^2 + k + 2).

%F Equals Sum_{k>=0} (-1)^A000120(k)/(A029931(k)+1) (Borwein and Borwein, 1992). (End)

%e 0.3684125359314336523213165973278510150142413039288199683036158...

%p evalf(8*sqrt(3/23)*Pi*sinh(sqrt(23)*Pi/6)/(2*cosh(sqrt(23)*Pi/3)-1), 123);

%p evalf(Sum((-1)^n/((3*n-1)*n/2 + 1), n=-infinity..infinity), 123);

%t RealDigits[N[8*Sqrt[3/23]*Pi*Sinh[Sqrt[23]*Pi/6] / (2*Cosh[Sqrt[23]*Pi/3]-1),120]][[1]]

%o (PARI) 8*Pi*sqrt(3/23) * sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3) - 1) \\ _Michel Marcus_, Nov 28 2018

%Y Cf. A000120, A010815, A029931, A242168, A258229, A258230, A258408.

%Y Cf. A258406 (m=2), A258407 (m=3), A258404 (m=4), A258405 (m=5).

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, May 24 2015