A258406 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^2 dx.

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

Offset

0,1

Links

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

Vaclav Kotesovec, The integration of q-series

Formula

Equals Sum_{n>=0} Sum_{k=0..n} 8*(n+1)*(-1)^n / ((n^2 - 2*k^2 + 2*k*n + n + 2) * (n^2 - 2*k^2 + 2*k*n + 5*n + 6)).

Equals Sum_{n>=0} Sum_{j=-floor(n/2)..floor(n/2)} (-1)^(n+j) / (n*(n+1)/2 - j*(3*j-1)/2 + 1).

Example

0.2538740823782760029885088938163329123847636343193313514756067...

Maple

evalf(Sum(Sum(8*(n+1)*(-1)^n / ((n^2 - 2*k^2 + 2*k*n + n + 2) * (n^2 - 2*k^2 + 2*k*n + 5*n + 6)), k=0..n), n=0..infinity), 120);

Mathematica

RealDigits[NIntegrate[QPochhammer[x]^2, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

Crossrefs

Cf. A002107, A258232, A258407, A258404, A258405.

Sequence in context: A257271 A257280 A050171 * A136189 A309557 A292965

Adjacent sequences: A258403 A258404 A258405 * A258407 A258408 A258409

Keyword

nonn,cons

Author

Vaclav Kotesovec, May 29 2015

Status

approved