A100221 Decimal expansion of Product_{k>=1} (1-1/4^k).
6, 8, 8, 5, 3, 7, 5, 3, 7, 1, 2, 0, 3, 3, 9, 7, 1, 5, 4, 5, 6, 5, 1, 4, 3, 5, 7, 2, 9, 3, 5, 0, 8, 1, 8, 4, 6, 7, 5, 5, 4, 9, 8, 1, 9, 3, 7, 8, 3, 3, 5, 7, 3, 5, 3, 4, 0, 1, 5, 7, 2, 3, 2, 5, 7, 7, 5, 3, 3, 1, 9, 8, 4, 5, 0, 7, 9, 8, 6, 7, 5, 1, 2, 4, 8, 0, 3, 3, 4, 6, 0, 4, 8, 1, 4, 2, 8, 8, 7, 9, 0, 5
Offset: 0
Examples
0.68853753712033971545651435729350818467554981937833...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1200
- Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.
- Eric Weisstein's World of Mathematics, Infinite Product.
- Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
Crossrefs
Programs
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Mathematica
EllipticThetaPrime[1, 0, 1/2]^(1/3)/2^(1/4) N[QPochhammer[1/4]] (* G. C. Greubel, Nov 30 2015 *) RealDigits[Fold[Times,1-1/4^Range[1000]],10,110][[1]] (* Harvey P. Dale, Sep 27 2024 *)
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PARI
prodinf(x=1, 1-1/4^x) \\ Altug Alkan, Dec 01 2015
Formula
Equals exp(-Sum_{k>0} sigma_1(k)/(k*4^k)) where sigma_1() is A000203(). - Hieronymus Fischer, Aug 07 2007
Equals (1/4; 1/4){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(Pi/log(2)) * exp(log(2)/12 - Pi^2/(12*log(2))) * Product_{k>=1} (1 - exp(-2*k*Pi^2/log(2))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027637(n). (End)