A241991 Decimal expansion of the infinite product for n >= 1 of (1+1/n^2)^(n^2)/e.

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

Offset

0,1

Links

Table of n, a(n) for n=0..102.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 3.

S. R. Holcombe, A product representation for Pi, arXiv:1204.2451 [math.NT], 2012.

Formula

Equals exp(1/2 + 2*Pi/3 - 1/(2*Pi^2)*zeta(3) + 1/(2*Pi^2)*Li_3(exp(-2*Pi)) + (1/Pi)*Li_2(exp(-2*Pi)))/(2*sinh(Pi)), where Li_n is the n-th polylogarithm.

Example

0.54578183883398708252349039725565877403368791329804393276759526235...

Maple

evalf(product((1+1/n^2)^(n^2)/exp(1), n=1..infinity), 120) # Vaclav Kotesovec, Sep 17 2014

Mathematica

p = Exp[1/2 + 2*Pi/3 - 1/(2*Pi^2)*Zeta[3] + 1/(2*Pi^2)*PolyLog[3, Exp[-2*Pi]] + (1/Pi)*PolyLog[2, Exp[-2*Pi]]]/(2*Sinh[Pi]); RealDigits[p, 10, 103] // First (* corrected by Eric Rowland, May 31 20122 *)

Prog

(PARI) exp(1/2 + 2*Pi/3 - zeta(3)/(2*Pi^2) + polylog(3, exp(-2*Pi))/(2*Pi^2) + polylog(2, exp(-2*Pi))/Pi)/2/sinh(Pi) \\ Charles R Greathouse IV, Aug 27 2014
(Python)
from mpmath import *
mp.dps=104
C = exp(1/2 + 2*pi/3 - zeta(3)/(2*pi**2) + polylog(3, exp(-2*pi))/(2*pi**2) + polylog(2, exp(-2*pi))/pi)/2/sinh(pi)
print([int(n) for n in list(str(C)[2:-1])]) # Indranil Ghosh, Jul 03 2017

Crossrefs

Cf. A240984, A247444

Sequence in context: A119807 A254181 A245018 * A184306 A176317 A092426

Adjacent sequences: A241988 A241989 A241990 * A241992 A241993 A241994

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Aug 11 2014

Status

approved