A047788 Numerators of Glaisher's I-numbers.

1, 1, 1, 7, 809, 1847, 55601, 6921461, 126235201, 8806171927, 2288629046003, 80348736972167, 10111159088668001, 40453941942593304589, 258227002122139705201, 51215766794507248883047, 34747165199239302488636803, 2962605017328303351107945687

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac

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

Examples

			1/2, 1/3, 1, 7, 809/9, 1847, 55601, 6921461/3, ...
		

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.
• Index entries for sequences related to Glaisher's numbers

Crossrefs

Cf. A047789, A002111.

Cf. A102283.

Programs

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

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 *)
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 *)

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 */

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

Formula

E.g.f. for (-1)^n*I(n) is (3/2)/(1 + 2*cosh(x)).