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 A047788 Numerators of Glaisher's I-numbers. 6
 1, 1, 1, 7, 809, 1847, 55601, 6921461, 126235201, 8806171927, 2288629046003, 80348736972167, 10111159088668001, 40453941942593304589, 258227002122139705201, 51215766794507248883047, 34747165199239302488636803, 2962605017328303351107945687 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Conjecture: L(2n+1, chi3) = a(n)/A047789(n) * (2*Pi)^(2n+1)/((2n)!*3^(2n+3/2)), where L(s, chi3) = Sum_{k>=1} Legendre(k,3)/k^s = Sum_{k>=1} A102283(k)/k^s is the Dirichlet L-function for the non-principal character modulo 3. - Jianing Song, Nov 17 2019 LINKS Robert Israel, Table of n, a(n) for n = 0..255 J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235. FORMULA E.g.f. for (-1)^n*I(n) is (3/2)/(1 + 2*cosh(x)). EXAMPLE 1/2, 1/3, 1, 7, 809/9, 1847, 55601, 6921461/3, ... MAPLE S:= series(3/(2+4*cos(x)), x, 101): seq(numer(coeff(S, x, 2*j)*(2*j)!), j=0..50); # Robert Israel, Aug 14 2018 MATHEMATICA terms = 20; CoefficientList[(3/2)/(1+Exp[x]+Exp[-x]) + O[x]^(2terms), x]* Range[0, 2terms-2]! // Abs // Numerator // DeleteCases[#, 0]& (* Jean-François Alcover, Feb 28 2019 *) PROG (PARI) a(n)=if(n<1, (n==0), n*=2; numerator(n!* polcoeff(3/(2+4*cos(x+O(x^n) )), n))) /* Michael Somos, Feb 26 2004 */ (Magma) m:=60; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 3/(2*(1+2*Cosh(x))) )); [Numerator((-1)^(n+1)*Factorial(2*n-2)* b[2*n-1]): n in [1..Floor((m-2)/2)]]; // G. C. Greubel, May 17 2019 (Sage) [numerator( (-1)^n*factorial(2*n)*( 3/(2*(1+2*cosh(x))) ).series(x, 2*n+2).list()[2*n]) for n in (0..30)] # G. C. Greubel, May 17 2019 CROSSREFS Cf. A047789, A002111. Cf. A102283. Sequence in context: A342834 A278438 A279120 * A251698 A203694 A269896 Adjacent sequences: A047785 A047786 A047787 * A047789 A047790 A047791 KEYWORD nonn,frac AUTHOR STATUS approved

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Last modified March 30 06:16 EDT 2023. Contains 361606 sequences. (Running on oeis4.)