A081855 - OEIS

%I #37 Oct 27 2024 09:23:51

%S 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,

%T 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,

%U 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

%N Decimal expansion of Gamma''(1).

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

%D Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 179.

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.2, p. 31.

%H G. C. Greubel, <a href="/A081855/b081855.txt">Table of n, a(n) for n = 1..10000</a>

%H Tom M. Apostol, <a href="https://doi.org/10.1090/S0025-5718-1985-0771044-5">Formulas for higher derivatives of the Riemann zeta function</a>, Mathematics of Computation 44 (1985), p. 223-232.

%F 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.

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

%t EulerGamma^2 + Zeta[2] // RealDigits[#, 10, 105] & // First (* _Jean-François Alcover_, Apr 29 2013 *)

%t RealDigits[ Integrate[ Exp[-x]*Log[x]^2, {x, 0, Infinity}], 10, 111][[1]] (* _Robert G. Wilson v_, Aug 18 2017 *)

%o (PARI) Euler^2+zeta(2) \\ _Charles R Greathouse IV_, Aug 18 2017

%o (PARI) intnum(x=0,[oo,1],exp(-x)*log(x)^2) \\ _Charles R Greathouse IV_, Aug 18 2017

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); EulerGamma(R)^2 + Evaluate(L,2); // _G. C. Greubel_, Aug 29 2018

%Y Cf. A001620, A013661, A155969, A261509.

%K cons,nonn

%O 1,2

%A _Benoit Cloitre_, Apr 11 2003