A385960 -id:A385960 - cp's OEIS frontend

A385961 Decimal expansion of the value of the coefficient [x^3] Gamma(x).

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

Offset: 0

R. J. Mathar, Jul 13 2025

The Laurent series Gamma(x) = 1/x + Sum_{i>=0} a_i x^i starts with a_0 = -gamma = -A001620, a_1 = A090998 . a_3 = 0.9817280868.. is here.

			0.981728086834400187336380294021850...

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

Cf. A090998 [x^1], A385960 [x^2], A385962 [x^4].

(gamma^4+6*gamma^2*Zeta(2)+8*gamma*Zeta(3)+3*Zeta(2)^2+6*Zeta(4))/24 ; evalf(%) ;

Equals (gamma^4 +6*gamma^2*zeta(2) +8*gamma*zeta(3) +3*zeta(2)^2 +6*zeta(4))/24 , gamma = A001620, zeta(2) = A013661, zeta(3)=A002117, zeta(4) = A013662.

A385962 Decimal expansion of the absolute value of the coefficient [x^4] Gamma(x).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 13 2025

The Laurent series Gamma(x) = 1/x + Sum_{i>=0} a_i x^i starts with a_0 = -gamma = -A001620, a_1 = A090998 . a_4 = -0.98199506.. , absolute value here.

			0.9819950689031452021047014137..

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

Cf. A090998 [x^1], A385960 [x^2], A385961 [x^3].

(gamma^5 +10*gamma^3*Zeta(2) +20*gamma^2*Zeta(3) +15*(Zeta(2)^2+2*Zeta(4))*gamma +20*Zeta(2)*Zeta(3) +24*Zeta(5))/120 ; evalf(%) ;

Equals (gamma^5 +10*gamma^3*zeta(2) +20*gamma^2*zeta(3) +15*(zeta(2)^2+2*zeta(4))*gamma +20*zeta(2)*zeta(3) +24*zeta(5))/120 , gamma = A001620, zeta(2) = A013661, zeta(3)=A002117, zeta(4) = A013662, zeta(5) = A013663.

cp's OEIS Frontend

A385961 Decimal expansion of the value of the coefficient [x^3] Gamma(x).

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