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

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, 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, 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

Offset: 1

Author

William J. Keith, May 05 2014

Keywords

• cons
• nonn
• nice

Comments

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).

Examples

			1.2883008886739212301809014939309634442258738...
		

Links

• Vaclav Kotesovec, The integration of q-series
• Vaclav Kotesovec, Graph of the area below a curve

Crossrefs

Cf. A010815, A143441, A258232.

Programs

Maple

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

Mathematica

NIntegrate[QPochhammer[q, q], {q, -1, 1}, WorkingPrecision -> 45]
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 *)

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.))
intnum(q=-.99999,.99999,eta2(q)) \\ Bill Allombert, May 06 2014

Formula

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

Extensions

More digits from Vaclav Kotesovec, Jun 02 2015