A377023 - OEIS

%I #13 Oct 16 2024 16:35:07

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

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

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

%N Decimal expansion of the asymptotic constant of the product of binomial coefficients in a row of Pascal's triangle.

%C The asymptotic product of binomial coefficients in the n-th row of Pascal's triangle as n goes to infinity provides an asymptotic constant C. This constant must lie in the interval [0.590727...,0.631618...), where the interval is derived from asymptotic products of binomial coefficients over the rows. Indeed, the constant C can also be derived from a limiting case of the latter products (see Kellner 2024).

%C The constant C is involved with a certain constant F(1) = A213080. The constants F(1), 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. By a divergent series expansion, it follows that F(1) lies in the interval (1.0457...,1.0492...) (see Kellner 2009 and 2024).

%H Bernd C. Kellner, <a href="/A377023/b377023.txt">Table of n, a(n) for n = 0..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 Let A = A074962 denote the Glaisher-Kinkelin constant.

%F Equals 1/(A213080*(2*Pi)^(1/4)).

%F Equals A^2/(exp(1/12)*(2*Pi)^(1/2)).

%F Equals exp(1/12-2*zeta'(-1))/(2*Pi)^(1/2).

%e 0.60364867603601031967070211804205268306704463040701700740585803621917783756033...

%p exp(1/12-2*Zeta(1, -1))/(2*Pi)^(1/2); evalf(%, 100);

%t RealDigits[Glaisher^2/(Exp[1/12] (2 Pi)^(1/2)), 10, 100][[1]]

%o (Sage)

%o import mpmath

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

%o mpmath.exp(1/12-2*mpmath.zeta(-1, 1, 1))/(2*pi)^(1/2)

%o (PARI)

%o default(realprecision, 100);

%o exp(1/12-2*zeta'(-1))/(2*Pi)^(1/2)

%Y Cf. A001620, A002117, A074962, A213080, A377024.

%K nonn,cons

%O 0,1

%A _Bernd C. Kellner_, Oct 13 2024