cp's OEIS Frontend

A242168 Decimal expansion of the integral of the q-Pochhammer symbol (reciprocal of the partition function) over the real interval -1 to 1.

%I A242168 #24 Jun 05 2015 04:15:33
%S A242168 1,2,8,8,3,0,0,8,8,8,6,7,3,9,2,1,2,3,0,1,8,0,9,0,1,4,9,3,9,3,0,9,6,3,
%T A242168 4,4,4,2,2,5,8,7,3,8,0,7,1,3,8,7,9,6,1,9,5,0,3,2,0,1,4,9,4,2,6,9,8,6,
%U A242168 4,4,2,4,1,8,5,2,0,4,9,7,8,8,7,6,8,2,0,9,3,4,4,4,4,1,1,1,3,3,9,8,1,3,6,3,3
%N A242168 Decimal expansion of the integral of the q-Pochhammer symbol (reciprocal of the partition function) over the real interval -1 to 1.
%C A242168 As a function, the q-Pochhammer symbol is an irregularly left-skewed bell curve. It has limiting value 0 at -1 and 1, and its maximum is at -0.411248... (decimal value given by A143441).
%H A242168 Vaclav Kotesovec, <a href="http://oeis.org/A258232/a258232_2.pdf">The integration of q-series</a>
%H A242168 Vaclav Kotesovec, <a href="/A242168/a242168.jpg">Graph of the area below a curve</a>
%F A242168 Equals 4*sqrt(3/23)*Pi * (2*sinh(sqrt(23)*Pi/6) + sqrt(2)*sinh(sqrt(23)*Pi/4)) / (2*cosh(sqrt(23)*Pi/3)-1). - _Vaclav Kotesovec_, Jun 02 2015
%e A242168 1.2883008886739212301809014939309634442258738...
%p A242168 evalf(4*sqrt(3/23)*Pi * (2*sinh(sqrt(23)*Pi/6) + sqrt(2)*sinh(sqrt(23)*Pi/4)) / (2*cosh(sqrt(23)*Pi/3)-1), 120); # _Vaclav Kotesovec_, Jun 02 2015
%t A242168 NIntegrate[QPochhammer[q, q], {q, -1, 1}, WorkingPrecision -> 45]
%t A242168 RealDigits[4*Sqrt[3/23]*Pi*(2*Sinh[Sqrt[23]*Pi/6] + Sqrt[2]*Sinh[Sqrt[23]*Pi/4]) / (2*Cosh[Sqrt[23]*Pi/3]-1), 10, 105][[1]] (* _Vaclav Kotesovec_, Jun 02 2015 *)
%o A242168 (PARI) eta2(q)=if(q==0,1,my(p=log(10^-38)/log(abs(q)),N=floor(sqrt(2*p/3)));sum(n=-N,N,(-1)^n*q^((3*n^2-n)/2),0.))
%o A242168 intnum(q=-.99999,.99999,eta2(q)) \\ Bill Allombert, May 06 2014
%Y A242168 Cf. A010815, A143441, A258232.
%K A242168 cons,nonn,nice
%O A242168 1,2
%A A242168 _William J. Keith_, May 05 2014
%E A242168 More digits from _Vaclav Kotesovec_, Jun 02 2015