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

1,2

Comments

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

References

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.

Links

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.

Formula

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.

Example

1.978111990655945110790791303001269415878367... [corrected by Georg Fischer, Jul 29 2021]

Mathematica

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 *)

Prog

(PARI) Euler^2+zeta(2) \\ Charles R Greathouse IV, Aug 18 2017
(PARI) intnum(x=0, [oo, 1], exp(-x)*log(x)^2) \\ Charles R Greathouse IV, Aug 18 2017
(Magma) SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); EulerGamma(R)^2 + Evaluate(L, 2); // G. C. Greubel, Aug 29 2018

Crossrefs

Cf. A001620, A013661, A155969, A261509.

Sequence in context: A232128 A336081 A086278 * A019887 A163931 A277774

Adjacent sequences: A081852 A081853 A081854 * A081856 A081857 A081858

Keyword

cons,nonn

Author

Benoit Cloitre, Apr 11 2003

Status

approved