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

Author

R. J. Mathar, Jul 13 2025

Keywords

• nonn
• cons

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} .

Examples

			0.9074790760808862890165601673...
		

Links

• I. S. Gradsteyn, 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)

Crossrefs

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

Programs

Maple

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

PARI

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

Formula

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