Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #14 Oct 10 2023 07:53:36
%S 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,
%T 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,
%U 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
%N Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^2 dx.
%H Vaclav Kotesovec, <a href="http://oeis.org/A258232/a258232_2.pdf">The integration of q-series</a>
%F 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)).
%F 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).
%e 0.2538740823782760029885088938163329123847636343193313514756067...
%p 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);
%t RealDigits[NIntegrate[QPochhammer[x]^2, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* _Vaclav Kotesovec_, Oct 10 2023 *)
%Y Cf. A002107, A258232, A258407, A258404, A258405.
%K nonn,cons
%O 0,1
%A _Vaclav Kotesovec_, May 29 2015