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 A082633 Decimal expansion of the 1st Stieltjes constant gamma_1 (negated). 64
 0, 7, 2, 8, 1, 5, 8, 4, 5, 4, 8, 3, 6, 7, 6, 7, 2, 4, 8, 6, 0, 5, 8, 6, 3, 7, 5, 8, 7, 4, 9, 0, 1, 3, 1, 9, 1, 3, 7, 7, 3, 6, 3, 3, 8, 3, 3, 4, 3, 3, 7, 9, 5, 2, 5, 9, 9, 0, 0, 6, 5, 5, 9, 7, 4, 1, 4, 0, 1, 4, 3, 3, 5, 7, 1, 5, 1, 1, 4, 8, 4, 8, 7, 8, 0, 8, 6, 9, 2, 8, 2, 4, 4, 8, 4, 4, 0, 1, 4, 6, 0, 4 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 166. LINKS G. C. Greubel, Table of n, a(n) for n = 0..2500 D. Andrica and E. J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanta, Volume XXII (2014) fascicola 1. Eric Weisstein's World of Mathematics, Stieltjes Constants Wikipedia, Stieltjes constants FORMULA Equals lim_{y->infinity} y*(Im(zeta(1+i/y))+y). Equals lim_{n->infinity} (((log(n))^2)/2 - sum((log(k))/k, k=2..n)). - Warut Roonguthai, Aug 04 2005 Also equals integral_[0..infinity] (coth(Pi*x)-1)*(x*log(1+x^2)-2*arctan(x))/(2*(1+x^2)) dx. - Jean-François Alcover, Jan 28 2015 Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma_1 = -(Pi/2)*Integral_{0..infinity} (a^2 - b^2)/c^2. The general case is for n >= 0 (which includes Euler's gamma as gamma_0) gamma_n = -(Pi/(n+1))* Integral_{0..infinity} sigma(n+1)/c^2, where sigma(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*b^(2*k)*a^(n-2*k). - Peter Luschny, Apr 19 2018 EXAMPLE -0.0728158454836767248605863758749... MAPLE evalf(gamma(1)) ; # R. J. Mathar, Sep 15 2013 MATHEMATICA Prepend[RealDigits[c=N[StieltjesGamma[1], 120], 10][[1]], 0] N[EulerGamma^2 - Residue[Zeta[s]^3, {s, 1}]/3, 100] (* Vaclav Kotesovec, Jan 07 2017 *) PROG (PARI) intnum(x=0, oo, (1/tanh(Pi*x)-1)*(x*log(1+x^2)-2*atan(x))/(2*(1+x^2))) \\ Charles R Greathouse IV, Mar 10 2016 CROSSREFS Cf. A001620, A086279, A086280, A086281, A086282, A183141, A183167, A183206, A184853, A184854. Sequence in context: A253383 A010506 A197845 * A121239 A201322 A093753 Adjacent sequences:  A082630 A082631 A082632 * A082634 A082635 A082636 KEYWORD cons,nonn AUTHOR Benoit Cloitre, May 24 2003 EXTENSIONS More terms from Eric W. Weisstein, Jul 14 2003 STATUS approved

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Last modified June 7 05:20 EDT 2020. Contains 334837 sequences. (Running on oeis4.)