A261509 -id:A261509 - cp's OEIS frontend

A081855 Decimal expansion of Gamma''(1).

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

Offset: 1

Benoit Cloitre, Apr 11 2003

Also the decimal expansion of the Integral_{x>=0} exp(-x)*(log(x))^2 dx. - Robert G. Wilson v, Aug 18 2017

			1.978111990655945110790791303001269415878367... [corrected by _Georg Fischer_, Jul 29 2021]

Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 179.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.2, p. 31.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Tom M. Apostol, Formulas for higher derivatives of the Riemann zeta function, Mathematics of Computation 44 (1985), p. 223-232.

Cf. A001620, A013661, A155969, A261509.

SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); EulerGamma(R)^2 + Evaluate(L,2); // G. C. Greubel, Aug 29 2018

EulerGamma^2 + Zeta[2] // RealDigits[#, 10, 105] & // First (* Jean-François Alcover, Apr 29 2013 *)
RealDigits[ Integrate[ Exp[-x]*Log[x]^2, {x, 0, Infinity}], 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2017 *)

Euler^2+zeta(2) \\ Charles R Greathouse IV, Aug 18 2017

intnum(x=0,[oo,1],exp(-x)*log(x)^2) \\ Charles R Greathouse IV, Aug 18 2017

The second derivative of Gamma(x) at x=1 is Gamma^2+zeta(2) = 1.97811199... where Gamma is the Euler constant and zeta(2) = Pi^2/6.

A291486 Decimal expansion of Gamma''''(1).

Original entry on oeis.org

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

Offset: 2

Robert G. Wilson v, Aug 24 2017

			23.56147408402560449607312705652442040865376831336316996971897893425256...

G. C. Greubel, Table of n, a(n) for n = 2..10000

Cf. A000796 (Pi), A001620 (EulerGamma), A002117 (zeta(3)), A081855, A261509.

SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); EulerGamma(R)^4 + EulerGamma(R)^2*Pi(R)^2 + 8*EulerGamma(R)*Evaluate(L,3) + 3*Pi(R)^4/20; // G. C. Greubel, Sep 07 2018

c:= subs(x=1.0, diff(GAMMA(x), x$4)):
evalf(c, 120);  # Alois P. Heinz, Jul 01 2023

Mathematica
```
RealDigits[Gamma''''[1], 10, 111][[1]]
```

default(realprecision, 100); Euler^4 + Euler^2*Pi^2 + 8*Euler*zeta(3) + 3*Pi^4/20 \\ G. C. Greubel, Sep 07 2018

Equals EulerGamma^4 + EulerGamma^2*Pi^2 + 8*EulerGamma*Zeta(3) + 3*Pi^4/20.

Equals Integral_{x=0..oo} exp(-x)*log(x)^4 dx. - Amiram Eldar, Aug 06 2020

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A081855 Decimal expansion of Gamma''(1).

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A291486 Decimal expansion of Gamma''''(1).

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