A258408 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^(2*k)) dx.

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

Offset

0,1

Comments

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

if 0<m<24: 8*sqrt(3)*Pi * sinh(Pi/6*sqrt(24/m-1)) /

(sqrt((24-m)*m) * (2*cosh(Pi/3*sqrt(24/m-1))-1))

if m = 24: Pi^2/(6*sqrt(3)) = A258414

if m > 24: 8*sqrt(3)*Pi*sin(Pi/6*sqrt(1-24/m)) /

(sqrt((m-24)*m) * (2*cos(Pi/3*sqrt(1-24/m))-1)).

Integral_{x=0..1} Product_{k=1..n} (1+x^(m*k)) dx, where m >= 1, is asymptotic to 2*(m+1)^(n+1)/(m*n^2).

Integral_{x=-1..1} Product_{k>=1} (1-x^(2*k)) dx = 8*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3)-1) = 1.154664240367959678... . - Vaclav Kotesovec, Jun 02 2015

Links

Table of n, a(n) for n=0..105.

Vaclav Kotesovec, The integration of q-series

Formula

Equals 4*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3) - 1).

Example

0.5773321201839797055525469620159041550801193138356349245589088...

Maple

evalf(4*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3) - 1), 120);

Mathematica

RealDigits[4*Pi*Sqrt[3/11]*Sinh[Sqrt[11]*Pi/6] / (2*Cosh[Sqrt[11]*Pi/3] - 1), 10, 120][[1]]

Crossrefs

Cf. A258232, A258406, A258414.

Sequence in context: A084823 A117034 A021638 * A210623 A020760 A225155

Adjacent sequences: A258405 A258406 A258407 * A258409 A258410 A258411

Keyword

nonn,cons

Author

Vaclav Kotesovec, May 29 2015

Status

approved