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A081855 Decimal expansion of Gamma''(1).

%I A081855 #37 Oct 27 2024 09:23:51
%S A081855 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 A081855 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 A081855 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 A081855 Decimal expansion of Gamma''(1).
%C A081855 Also the decimal expansion of the Integral_{x>=0} exp(-x)*(log(x))^2  dx. - _Robert G. Wilson v_, Aug 18 2017
%D A081855 Bruce C. Berndt, Ramanujan's notebooks Part II, Springer, p. 179.
%D A081855 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.2, p. 31.
%H A081855 G. C. Greubel, <a href="/A081855/b081855.txt">Table of n, a(n) for n = 1..10000</a>
%H A081855 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 A081855 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 A081855 1.978111990655945110790791303001269415878367... [corrected by _Georg Fischer_, Jul 29 2021]
%t A081855 EulerGamma^2 + Zeta[2] // RealDigits[#, 10, 105] & // First (* _Jean-François Alcover_, Apr 29 2013 *)
%t A081855 RealDigits[ Integrate[ Exp[-x]*Log[x]^2, {x, 0, Infinity}], 10, 111][[1]] (* _Robert G. Wilson v_, Aug 18 2017 *)
%o A081855 (PARI) Euler^2+zeta(2) \\ _Charles R Greathouse IV_, Aug 18 2017
%o A081855 (PARI) intnum(x=0,[oo,1],exp(-x)*log(x)^2) \\ _Charles R Greathouse IV_, Aug 18 2017
%o A081855 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); L:=RiemannZeta(); EulerGamma(R)^2 + Evaluate(L,2); // _G. C. Greubel_, Aug 29 2018
%Y A081855 Cf. A001620, A013661, A155969, A261509.
%K A081855 cons,nonn
%O A081855 1,2
%A A081855 _Benoit Cloitre_, Apr 11 2003