A258407 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^3 dx.

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

Offset: 0

Author

Vaclav Kotesovec, May 29 2015

Keywords

• nonn
• cons
• changed

Comments

In general, Integral_{x=0..1} Product_{k>=1} (1-x^(m*k))^3 dx = Sum_{n>=0} (-1)^n * (2*n+1) / (m*n*(n+1)/2 + 1) is equal to

if 0

if m = 8: Pi/4

if m > 8: 2*Pi / (m * cos((Pi/2)*sqrt(1-8/m)))

Special values: m=4: Pi/(2*cosh(Pi/2)), m=9: 4*Pi/(9*sqrt(3)).

---

Integral_{x=-1..1} Product_{k>=1} (1-x^k)^3 dx = 2*Pi*(1 + sqrt(2) * cosh(sqrt(7)*Pi/4)) / cosh(sqrt(7)*Pi/2) = 1.32639350417409769439126... . - Vaclav Kotesovec, Jun 02 2015

Examples

			0.1968806153145889753533513584769666829667343178391757586093357...
		

Links

• Vaclav Kotesovec, The integration of q-series

Crossrefs

Cf. A010816, A258232, A258406, A258404, A258405.

Programs

Maple

evalf(2*Pi/cosh(sqrt(7)*Pi/2), 120);
evalf(Sum((-1)^n * (2*n+1) / (n*(n+1)/2 + 1), n=0..infinity), 120);

Mathematica

RealDigits[2*Pi*Sech[(Sqrt[7]*Pi)/2],10,105][[1]]

PARI

2*Pi/cosh((sqrt(7)*Pi)/2) \\ Stefano Spezia, Aug 23 2025

Formula

Equals 2*Pi/cosh(sqrt(7)*Pi/2).

Equals Sum_{n>=0} (-1)^n * (2*n+1) / (n*(n+1)/2 + 1).

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.