A385960 Decimal expansion of the absolute value of the coefficient [x^2] Gamma(x).

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

Offset

0,1

Comments

The Laurent series Gamma(x) = 1/x + Sum_{i>=0} a_i * x^i starts with a_0 = -gamma = -A001620, a_1 = A090998 , and a_2 = -0.90747907... , absolute value here. Recurrence (i+1)*a_i = -gamma *a_{i-1} + Sum_{k=2..i+1} (-1)^k*zeta(k)*a_{i-k}.

Links

Table of n, a(n) for n=0..103.

I. S. Gradsteyn and I. M. Ryzhik, Tables of Series and Products, Academic Press (2014), 8.321.1.

R. J. Mathar, Erratum to Exercise A4.2 in "An Introduction to the Theory of the Riemann Zeta Function", viXra:2507.0094 (2025).

Formula

Equals (gamma^3 + 3*gamma*zeta(2) + 2*zeta(3))/6 , gamma = A001620, zeta(2) = A013661, zeta(3) = A002117.

Example

0.9074790760808862890165601673...

Maple

(gamma^3+3*gamma*Zeta(2)+2*Zeta(3))/6 ; evalf(%) ;

Mathematica

RealDigits[SeriesCoefficient[Gamma[x], {x, 0, 2}], 10, 120][[1]] (* Amiram Eldar, Apr 14 2026 *)

Prog

(PARI) polcoef(gamma(x), 2) \\ Michel Marcus, Jul 13 2025

Crossrefs

Cf. A090998 [x^1], A385961 [x^3], A385962 [x^4].

Cf. A001620, A002117, A013661.

Sequence in context: A384283 A372830 A390151 * A381653 A388361 A309823

Adjacent sequences: A385957 A385958 A385959 * A385961 A385962 A385963

Keyword

nonn,cons

Author

R. J. Mathar, Jul 13 2025

Status

approved