A389381 Decimal expansion of Product_{n>=1} sqrt(2*Pi) * (m/e)^m / e^(1/(24*m)) / n! (where m=n+1/2), the cumulative error ratio of the first-order refinement of a variant of Stirling's approximation over the positive integers.

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

Offset

0,1

Comments

Main entry: A389380.

This is the approximation described in A277002/A277003 (see also A189048/A189049), truncated to the first term.

This variant is attributed to Burnside but was the first one Stirling discovered, then simplified into the standard approximation named after him. However, this has asymptotically half the error (and is an upper bound instead of lower).

Multiplying Stirling's nominal approximation by e^(1/(12*n)) and his original (this one) by e^(-1/(24*m)) flips their inequalities.

Note that while Stirling's nominal can be evaluated for n > 0, his original can be for n>-1/2; the product over all positive half-integer n is 2^(1/4) * A / e^(gamma/12+3/8), where A = A074962.

Links

Table of n, a(n) for n=0..99.

Jonathan M. Borwein and Robert M. Corless, Gamma and Factorial in the Monthly, Amer. Math. Monthly, 125(5) (2018), 400-424; arXiv version, arXiv:1703.05349 [math.HO], 2017.

Natalia L. Skirrow, Stirling's approximations

Formula

Equals sqrt(e*A/Pi)/(2*e^gamma)^(1/24), with gamma = A001620, A = A074962.

Equals Pi^(-1/2)*(exp(13 - gamma - 12*zeta'(-1))/2)^(1/24). - Peter Luschny, Oct 14 2025

Example

0.99908093296568954102773981568058...

Maple

split := r -> ListTools:-Reverse(convert(floor(r), base, 10)):
Digits := 120: Pi^(-1/2)*(exp(13 - gamma - 12*Zeta(1, -1))/2)^(1/24):
split(evalf(%) * 10^100);  # Peter Luschny, Oct 14 2025

Mathematica

RealDigits[Sqrt[E*Glaisher/Pi]/(2*E^EulerGamma)^(1/24), 10, 128][[1]]

Crossrefs

This is for the c=1/2 case of the general formula n! ~ sqrt(2*Pi)*(n+c)^(n+1/2)/e^(n+c); c=0 is A389380 and c=1 is A389382.

Euler-Maclaurin series for the error of log is A277002/A277003, following from A189048/A189049.

Cf. A001620, A074962.

Sequence in context: A339745 A346929 A346927 * A197149 A381397 A274031

Adjacent sequences: A389378 A389379 A389380 * A389382 A389383 A389384

Keyword

nonn,cons

Author

Natalia L. Skirrow, Sep 28 2025

Status

approved