A389380 Decimal expansion of Product_{n>=1} sqrt(2*Pi*n)*(n/e)^n*e^(1/(12*n)) / n!, the cumulative error ratio of the first-order refinement of Stirling's approximation over the positive integers.

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

Offset

1,4

Comments

The Laurent series for the error of Stirling's approximation of log(n!) (A046968/A046969) is given by the Euler-Maclaurin formula.

Refining Stirling's approximation, by including the first terms in these expansions, makes its logs' errors shrink like 1/n^3, so the sum of log-errors (i.e. log of product of errors) becomes convergent.

log-Gamma's Euler-Maclaurin series comes from applying Watson's lemma to Binet's first log-Gamma formula as a Laplace transform, so it is a rare case in which the converse of Watson's lemma holds; the closed form for the exact value can be found by summing the Laplace transforms and treating the resulting integral as the limit of a Mellin transform.

This is an approximation about n = 0; constants from that about n = -1/2, n = -1 are A389381, A389382. (The formula for Laurent expansions of log(n!) about n = -c was found in p.206 of the Hermite reference.)

The former has < 1/3rd the error, the latter has < 1/5th.

About 0 and -1, the first-order refinement turns a lower bound into an upper one, while about -1/2 it turns an upper into a lower one. These are special cases; the inequality's direction is not always flipped, and about some values the refinement is not an inequality at all (the error's sign is dependent on n).

Links

Table of n, a(n) for n=1..100.

Charles Hermite, Sur la fonction log Γ(a). J. Reine Angew. Math. 115, (1895): 201-208.

Gergő Nemes, Generalization of Binet's Gamma function formulas (2013). Integral Transforms and Special Functions 24, no. 8, 597-606.

Natalia L. Skirrow, Stirling's approximations

Formula

Equals e^((1+gamma)/12 - 2*zeta'(-1)) / tau^(1/4), where tau = A019692.

Equals A^2 / (e^((1-gamma)/12) * tau^(1/4)), where A = A074962.

Example

1.00281160470777211547901529908336...

Maple

exp((1 + gamma)/12 - 2*Zeta(1, -1)) / (2*Pi)^(1/4): evalf(%, 100); # Peter Luschny, Oct 02 2025

Mathematica

N[E^((1+EulerGamma)/12-2*Zeta'[-1])/(2*Pi)^(1/4), 128]

Crossrefs

Cf. A389381, A389382 (constants for shifted formulae; the latter is effectively this one).

Cf. A046968/A046969 (series whose first term produces this constant).

Cf. A019692, A074962.

Sequence in context: A200704 A257955 A024544 * A202539 A197287 A081882

Adjacent sequences: A389377 A389378 A389379 * A389381 A389382 A389383

Keyword

nonn,cons

Author

Natalia L. Skirrow, Sep 28 2025

Status

approved