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

0,2

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<m<8: (2*Pi / (m * cosh(Pi/2*sqrt(8/m-1)))

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)).

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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

Links

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

Vaclav Kotesovec, The integration of q-series

Formula

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

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

Example

0.1968806153145889753533513584769666829667343178391757586093357...

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]]

Crossrefs

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

Sequence in context: A198567 A021512 A154205 * A138500 A161484 A103985

Adjacent sequences: A258404 A258405 A258406 * A258408 A258409 A258410

Keyword

nonn,cons

Author

Vaclav Kotesovec, May 29 2015

Status

approved