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

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

Offset

1,3

Comments

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.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Michael D. Hirschhorn, On the asymptotic behavior of Product_{k=0..n} C(n,k), Fib. Q., 51 (2013), 163-173.

Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, arXiv:math/0604505 [math.NT], p. 7.

Formula

Let A denote the Glaisher-Kinkelin constant. Then

C = (exp(1)^(1/12)*(2*Pi)^(1/4))/A^2 = exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4).

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

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

Example

1.0463350667705031...

Maple

exp(2*Zeta(1, -1)-1/12)*(2*Pi)^(1/4); evalf(%, 100); # Peter Luschny, Jun 22 2012

Mathematica

RealDigits[(Exp[1]^(1/12) (2 Pi)^(1/4))/Glaisher^2, 10, 100][[1]] (*Peter Luschny, Jun 22 2012 *)

Prog

(Sage)
import mpmath
mpmath.mp.pretty=True; mpmath.mp.dps = 200 #precision
mpmath.exp(2*mpmath.zeta(-1, 1, 1)-1/12)*(2*pi)^(1/4) # Peter Luschny, Jun 22 2012
(PARI) exp(2*zeta'(-1)-1/12)*(2*Pi)^(1/4) \\ Charles R Greathouse IV, Dec 12 2013

Crossrefs

Cf. A074962, A000178, A084448, A241140, A272097.

Sequence in context: A338147 A338143 A016492 * A200365 A198121 A244020

Adjacent sequences: A213077 A213078 A213079 * A213081 A213082 A213083

Keyword

nonn,cons

Author

Michael David Hirschhorn, Jun 04 2012

Status

approved