A258405 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^5 dx.

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

Offset: 0

Author

Vaclav Kotesovec, May 29 2015

Keywords

• nonn
• cons

Examples

			0.137801070846554642845386131402193843084522541232625982683937003709248631...
		

Links

• Vaclav Kotesovec, The integration of q-series

Crossrefs

Cf. A000728, A258232, A258406, A258407, A258404, A258405.

Programs

Maple

evalf(Sum(Sum(2*Pi*(-1)^(h+m) / cosh(sqrt(7 - 4*h + 12*h^2 - 4*m + 12*m^2)*Pi/2), m=-infinity..infinity), h=-infinity..infinity), 120); # Vaclav Kotesovec, Dec 04 2015

Mathematica

nmax=200; p=1; q5=Table[PrintTemporary[n]; p=Expand[p*(1-x^n)^5]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]],{n,1,nmax}]; q5n=N[q5,1000]; Table[SequenceLimit[Take[q5n,j]],{j,Length[q5n]-100,Length[q5n],10}]
nterms = 40; N[Sum[Sum[2*Pi*(-1)^(h+m) / Cosh[Sqrt[7 - 4*h + 12*h^2 - 4*m + 12*m^2]*Pi/2], {m, -nterms, nterms}], {h, -nterms, nterms}], 100] (* Vaclav Kotesovec, Dec 04 2015 *)
RealDigits[NIntegrate[QPochhammer[x]^5, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

Formula

Sum_{m = -infinity..infinity} (Sum_{h = -infinity..infinity} (2*Pi*(-1)^(h+m) / cosh(sqrt(7 - 4*h + 12*h^2 - 4*m + 12*m^2)*Pi/2))). - Vaclav Kotesovec, Dec 04 2015