A385966 Decimal expansion of the value of the coefficient [x^5] 1/Gamma(x).

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

Offset

0,2

Comments

The Taylor series 1/Gamma(x) = Sum_{i>=1} c_i x^i starts with c_1 = 1, c_2 = gamma = A001620, c_3 = -0.655878... = -A070860 . c_5 = 0.166538... here.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, 6.1.34.

I. S. Gradsteyn, I. M. Ryzhik, Tables of Series and Products, Academic Press (2014) 8.321.2 gives recurrence.

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

Simon Plouffe, Table up to c_15, (2004)

J. W. Wrench, Concerning two series for the Gamma Function, Math. Comp. 22 (1968) 617-626, Table 5.

Formula

Equals (Pi^4 -60*Pi^2*gamma^2 +60*gamma^4 +480*gamma*zeta(3))/1440, gamma = A001620, zeta(3) = A002117, Pi = A000796.

Example

0.16653861138229148950170079510210523571...

Maple

(Pi^4-60*Pi^2*gamma^2+60*gamma^4+480*gamma*Zeta(3))/1440 ; evalf(%) ;

Mathematica

First[RealDigits[(Pi^4 - 60*Pi^2*#^2 + 60*#^4 + 480*#*Zeta[3])/1440 & [EulerGamma], 10, 100]] (* or *)
First[RealDigits[Module[{x}, SeriesCoefficient[1/Gamma[x], {x, 0, 5}]], 10, 100]] (* Paolo Xausa, Aug 08 2025 *)

Crossrefs

Cf. A001620 [x^2], A070860 [x^3], A385965 [x^4].

Sequence in context: A011188 A246184 A197478 * A260713 A101801 A350362

Adjacent sequences: A385963 A385964 A385965 * A385967 A385968 A385969

Keyword

nonn,cons

Author

R. J. Mathar, Jul 13 2025

Status

approved