A070860 -id:A070860 - cp's OEIS frontend

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

Offset: 0

R. J. Mathar, Jul 13 2025

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} .

			-0.042002635034095235529003934875....

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.

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

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

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 *)

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

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

Original entry on oeis.org

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

R. J. Mathar, Jul 13 2025

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.

			0.16653861138229148950170079510210523571...

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.

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

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

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 *)

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

cp's OEIS Frontend

A385965 Decimal expansion of the absolute value of the coefficient [x^4] 1/Gamma(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula