cp's OEIS Frontend

A213080 Decimal expansion of Product_{n>=1} n! /(sqrt(2*Pi*n) * (n/e)^n * (1+1/n)^(1/12)).

%I A213080 #55 Oct 16 2024 16:35:19
%S A213080 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 A213080 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 A213080 5,7,5,1,5,6,3,7,4,4,5,7,0,7,2,5,4,8,5,8
%N A213080 Decimal expansion of Product_{n>=1} n! /(sqrt(2*Pi*n) * (n/e)^n * (1+1/n)^(1/12)).
%C A213080 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.
%C A213080 From _Bernd C. Kellner_, Oct 13 2024: (Start)
%C A213080 It turns out that 1/C is not the complete asymptotic constant for the product of the binomial coefficients in a row of Pascal's triangle. A constant factor of (2*Pi)^(-1/4) was overlooked in the asymptotic expansion of that product given by Hirschhorn in 2013. The correct asymptotic constant is A377023.
%C A213080 However, the constant C equals the constant F(1) as introduced before in Kellner 2009. The constants F(1), F(2), ... occur in the same context of asymptotic constants related to asymptotic products of factorials as well as of binomial and multinomial coefficients. Moreover, the sequence (F(k))_{k >= 1} is strictly decreasing with limit 1. For example, for k >= 1 the asymptotic product Prod_{v >= 1} (k*v)! has the asymptotic constant F(k)*A^k*(2*Pi)^(1/4), where A = A074962 denotes the Glaisher-Kinkelin constant. Let gamma = A001620 be Euler's constant and Gamma(x) be the gamma function.
%C A213080 For k >= 1, the constants F(k) can be computed by an explicit formula and a divergent series expansion, as follows. We have log(F(k)) = (1/(12*k))*(1-log(k)) + (k/4)*log(2*Pi) - ((k^2+1)/k)*log(A) - Sum_{v=1..k-1} (v/k)*log(Gamma(v/k)) = gamma/(12*k) - t*zeta(3)/(360*k^3) with some t in (0,1), respectively.
%C A213080 It follows that log(F(1)) = 1/12 + log(2*Pi)/4 - 2*log(A) = gamma/12 - t*zeta(3)/360 with some t in (0,1), and so F(1) lies in the interval (1.0457...,1.0492...) (see Kellner 2009 and 2024). (End)
%H A213080 G. C. Greubel, <a href="/A213080/b213080.txt">Table of n, a(n) for n = 1..10000</a>
%H A213080 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 A213080 Bernd C. Kellner, <a href="https://doi.org/10.1515/INTEG.2009.009">On asymptotic constants related to products of Bernoulli numbers and factorials</a>, Integers 9 (2009), Article #A08, 83-106; <a href="https://www.emis.de/journals/INTEGERS/papers/j8/j8.Abstract.html">alternative link</a>; arXiv:<a href="https://arxiv.org/abs/math/0604505">0604505</a> [math.NT], 2006.
%H A213080 Bernd C. Kellner, <a href="https://doi.org/10.5281/zenodo.12167556">Asymptotic products of binomial and multinomial coefficients revisited</a>, Integers 24 (2024), Article #A59, 10 pp.; arXiv:<a href="https://arxiv.org/abs/2312.11369">2312.11369</a> [math.CO], 2023.
%F A213080 Equals (exp(1)^(1/12)*(2*Pi)^(1/4))/A^2 where A denotes the Glaisher-Kinkelin constant.
%F A213080 Equals exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4).
%F A213080 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 A213080 The sqrt of the constant equals 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 A213080 1.04633506677050318098095065697776037101974218113264442441587534042035751563744...
%p A213080 exp(2*Zeta(1,-1)-1/12)*(2*Pi)^(1/4); evalf(%,100); # _Peter Luschny_, Jun 22 2012
%t A213080 RealDigits[(Exp[1]^(1/12) (2 Pi)^(1/4))/Glaisher^2, 10, 100][[1]] (*_Peter Luschny_, Jun 22 2012 *)
%o A213080 (Sage)
%o A213080 import mpmath
%o A213080 mpmath.mp.pretty=True; mpmath.mp.dps = 200 #precision
%o A213080 mpmath.exp(2*mpmath.zeta(-1,1,1)-1/12)*(2*pi)^(1/4) # _Peter Luschny_, Jun 22 2012
%o A213080 (PARI) exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4) \\ _Charles R Greathouse IV_, Dec 12 2013
%Y A213080 Cf. A074962, A000178, A084448, A241140, A272097.
%Y A213080 Cf. A001620, A002117, A377023, A377024.
%K A213080 nonn,cons
%O A213080 1,3
%A A213080 _Michael David Hirschhorn_, Jun 04 2012