A258404 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^4 dx.

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

Offset

0,2

Links

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

Vaclav Kotesovec, The integration of q-series

Formula

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

Example

0.16182024229485656180261334985786534313068578288018990398...

Maple

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

Mathematica

nmax=200; p=1; q4=Table[PrintTemporary[n]; p=Expand[p*(1-x^n)^4]; Total[CoefficientList[p, x]/Range[1, Exponent[p, x]+1]], {n, 1, nmax}]; q4n=N[q4, 1000]; Table[SequenceLimit[Take[q4n, j]], {j, Length[q4n]-100, Length[q4n], 10}]
NSum[2*(-1)^m*Pi/Cosh[Sqrt[7 - 4*m + 12*m^2]*Pi/2], {m, -Infinity, Infinity}, WorkingPrecision -> 120, NSumTerms -> 100] (* Vaclav Kotesovec, Dec 04 2015 *)
RealDigits[NIntegrate[QPochhammer[x]^4, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

Prog

(PARI) default(realprecision, 93);
b(n) = cosh(sqrt(7 - 4*n + 12*n^2)*Pi/2);
2*Pi*(1/b(0) + sumalt(n=1, (-1)^n*(1/b(n) + 1/b(-n)))) \\ Gheorghe Coserea, Sep 26 2018

Crossrefs

Cf. A000727, A258232, A258406, A258407, A258404, A258405.

Sequence in context: A073228 A256853 A349905 * A244692 A248589 A288493

Adjacent sequences: A258401 A258402 A258403 * A258405 A258406 A258407

Keyword

nonn,cons

Author

Vaclav Kotesovec, May 29 2015

Extensions

More digits from Vaclav Kotesovec, Oct 10 2023

Status

approved