A047789 Denominators of Glaisher's I-numbers.

2, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 81, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• 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. A047788, A002111.

Programs

Maple

f:= n -> 3^padic:-ordp(2*n+1,3):
f(0):= 2:
map(f, [$0..200]); # Robert Israel, Aug 14 2018

Mathematica

a[0] = 2; a[n_] := 3^IntegerExponent[2n+1, 3];
Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Feb 27 2019 *)
a[0]:=2; a[n_]:=Denominator[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,0,100}] (* Detlef Meya, Sep 28 2024 *)

PARI

a(n)=if(n<1,2*(n==0),3^valuation(2*n+1,3)) /* Michael Somos, Feb 26 2004 */

PARI

a(n)=if(n<1,2*(n==0),n*=2;denominator(n!*polcoeff(3/(2+4*cos(x+O(x^n))),n))) /* Michael Somos, Feb 26 2004 */

Formula

From Robert Israel, Aug 14 2018: (Start)

For n >= 1, a(3*n) = a(3*n+2) = 1 and a(3*n+1) = 3*a(n).

G.f. g(x) satisfies g(x) = 3*x*g(x^3) + 2 - 3*x + (x^2+x^3)/(1-x^3). (End)

G.f.: 1 + Sum_{k>=0} (3^k*x^((5*3^k - 1)/2) + 3^k*x^((3^k - 1)/2))/(1 - x^(3^(k + 1))). - Miles Wilson, Dec 01 2024