A377024 - OEIS

%I #18 Oct 16 2024 16:35:11

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

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

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

%N Decimal expansion of the constant F(2) related to asymptotic products of factorials.

%C The constants F(1) = A213080, F(2), ... occur in the 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 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 It follows that log(F(2)) = 1/24 + log(2*Pi)/4 + (5/24)*log(2) - (5/2)*log(A) = gamma/24 - t*zeta(3)/2880 with some t in (0,1), and so F(2) lies in the interval (1.023914..., 1.024342...) (see Kellner 2009 and 2024).

%H Bernd C. Kellner, <a href="/A377024/b377024.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 Equals exp(1/24)*(2*Pi)^(1/4)*2^(5/24)/A^(5/2) where A = A074962.

%F Equals exp(-1/6+(5/2)*zeta'(-1))*(2*Pi)^(1/4)*2^(5/24).

%e 1.02393741163711840157795078258621780080376098043644005129469909513476924124007...

%p exp(-1/6+5/2*Zeta(1, -1))*(2*Pi)^(1/4)*2^(5/24); evalf(%, 100);

%t RealDigits[Exp[1/24] (2 Pi)^(1/4) 2^(5/24) / Glaisher^(5/2), 10, 100][[1]]

%o (Sage)

%o import mpmath

%o mpmath.mp.pretty = True; mpmath.mp.dps = 100

%o mpmath.exp(-1/6+5/2*mpmath.zeta(-1, 1, 1))*(2*pi)^(1/4)*2^(5/24)

%o (PARI)

%o default(realprecision, 100);

%o exp(-1/6+5/2*zeta'(-1))*(2*Pi)^(1/4)*2^(5/24)

%Y Cf. A001620, A002117, A074962, A213080, A377023.

%K nonn,cons

%O 1,3

%A _Bernd C. Kellner_, Oct 13 2024