|
%I #47 Jul 27 2018 04:34:30
%S 1,0,4,6,3,3,5,0,6,6,7,7,0,5,0,3,1,8,0,9,8,0,9,5,0,6,5,6,9,7,7,7,6,0,
%T 3,7,1,0,1,9,7,4,2,1,8,1,1,3,2,6,4,4,4,2,4,4,1,5,8,7,5,3,4,0,4,2,0,3,
%U 5,7,5,1,5,6,3,7,4,4,5,7,0,7,2,5,4,8,5,8
%N Decimal expansion of Product_{n>=1} n! /(sqrt(2*Pi*n) * (n/e)^n * (1+1/n)^(1/12)).
%C Just as Stirling's formula for the asymptotic expansion of n! involves the constant sqrt{2 Pi}, the asymptotic expansion of the product of all binomial coefficients in a row of Pascal's triangle involves a constant, the reciprocal of the constant C defined and evaluated here.
%H G. C. Greubel, <a href="/A213080/b213080.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael D. Hirschhorn, <a href="/A213080/a213080.pdf">On the asymptotic behavior of Product_{k=0..n} C(n,k)</a>, Fib. Q., 51 (2013), 163-173.
%H Bernd C. Kellner, <a href="http://arxiv.org/abs/math.NT/0604505">On asymptotic constants related to products of Bernoulli numbers and factorials</a>, arXiv:math/0604505 [math.NT], p. 7.
%F Let A denote the Glaisher-Kinkelin constant. Then
%F C = (exp(1)^(1/12)*(2*Pi)^(1/4))/A^2 = exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4).
%F A closely related constant is K = Product_{n>=1} (n!*(e/n)^(n+1/2))/ ((1+1/(n+1/2))^(1/12)*sqrt(2*Pi*e)) = (2^(1/6)*(3*e)^(1/12)*Pi^(1/4))/A^2 = exp(2*zeta'(-1)-1/12)*2^(1/6)*3^(1/12)*Pi^(1/4) = 1.082293504658977773529439... - _Peter Luschny_, Jun 22 2012
%F sqrt(C) = Limit_{n>=1} (Product_{k=1..n-1} k!) / f(n) where f(n) = (2*Pi)^(n/2-1/8)*exp(1/24-3/4*n^2)*n^(1/2*n^2-1/12). - _Peter Luschny_, Jun 23 2012
%e 1.0463350667705031...
%p exp(2*Zeta(1,-1)-1/12)*(2*Pi)^(1/4); evalf(%,100); # _Peter Luschny_, Jun 22 2012
%t RealDigits[(Exp[1]^(1/12) (2 Pi)^(1/4))/Glaisher^2, 10, 100][[1]] (*_Peter Luschny_, Jun 22 2012 *)
%o (Sage)
%o import mpmath
%o mpmath.mp.pretty=True; mpmath.mp.dps = 200 #precision
%o mpmath.exp(2*mpmath.zeta(-1,1,1)-1/12)*(2*pi)^(1/4) # _Peter Luschny_, Jun 22 2012
%o (PARI) exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4) \\ _Charles R Greathouse IV_, Dec 12 2013
%Y Cf. A074962, A000178, A084448, A241140, A272097.
%K nonn,cons
%O 1,3
%A _Michael David Hirschhorn_, Jun 04 2012
|
