A385965 - OEIS

A385965

Decimal expansion of the absolute value of the coefficient [x^4] 1/Gamma(x).

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

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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_4 = -0.04200263... , absolute value here. Recurrence (i-1)*c_i = gamma *c_{i-1} - Sum_{k=2..i-1} (-1)^k*zeta(k) * c_{i-k} .

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 (-4*zeta(3) +Pi^2*gamma -2*gamma^3)/12, gamma = A001620, zeta(3) = A002117, Pi = A000796.

EXAMPLE

-0.042002635034095235529003934875....

MAPLE

(4*Zeta(3)-Pi^2*gamma+2*gamma^3)/12 ; evalf(%) ;

MATHEMATICA

First[RealDigits[(Pi^2*EulerGamma - 2*EulerGamma^3 - 4*Zeta[3])/12, 10, 100, -1]] (* or *)

First[RealDigits[Module[{x}, SeriesCoefficient[1/Gamma[x], {x, 0, 4}]], 10, 100, -1]] (* Paolo Xausa, Aug 08 2025 *)

CROSSREFS

Cf. A001620 [x^2], A070860 [x^3], A385966 [x^5].

Sequence in context: A232833 A256269 A256279 * A363436 A277767 A107088

Adjacent sequences: A385962 A385963 A385964 * A385966 A385967 A385968

KEYWORD

nonn,cons

AUTHOR

R. J. Mathar, Jul 13 2025

STATUS

approved