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A047788 Numerators of Glaisher's I-numbers.

%I A047788 #37 Oct 23 2024 01:24:01
%S A047788 1,1,1,7,809,1847,55601,6921461,126235201,8806171927,2288629046003,
%T A047788 80348736972167,10111159088668001,40453941942593304589,
%U A047788 258227002122139705201,51215766794507248883047,34747165199239302488636803,2962605017328303351107945687
%N A047788 Numerators of Glaisher's I-numbers.
%C A047788 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
%H A047788 Robert Israel, <a href="/A047788/b047788.txt">Table of n, a(n) for n = 0..255</a>
%H A047788 J. W. L. Glaisher, <a href="https://doi.org/10.1112/plms/s1-31.1.216">On a set of coefficients analogous to the Eulerian numbers</a>, Proc. London Math. Soc., 31 (1899), 216-235.
%H A047788 <a href="/index/Ge#Glaisher">Index entries for sequences related to Glaisher's numbers</a>
%F A047788 E.g.f. for (-1)^n*I(n) is (3/2)/(1 + 2*cosh(x)).
%e A047788 1/2, 1/3, 1, 7, 809/9, 1847, 55601, 6921461/3, ...
%p A047788 S:= series(3/(2+4*cos(x)),x,101):
%p A047788 seq(numer(coeff(S,x,2*j)*(2*j)!),j=0..50); # _Robert Israel_, Aug 14 2018
%t A047788 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 *)
%t A047788 a[0]:=1; a[n_]:=Numerator[FunctionExpand[(PolyGamma[2*n, 1/3] + (3^(2*n+1)-1)*(2*n)!*Zeta[2*n+1]/2)*Sqrt[3]/(-2^(2*n)*Pi^(2*n+1))]]; Table[a[n], {n,0,17}] (* _Detlef Meya_, Sep 28 2024 *)
%o A047788 (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 */
%o A047788 (Magma) m:=60; R<x>:=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
%o A047788 (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
%Y A047788 Cf. A047789, A002111.
%Y A047788 Cf. A102283.
%K A047788 nonn,frac
%O A047788 0,4
%A A047788 _N. J. A. Sloane_