A091574 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type D_4.

5, 8, 15, 16, 25, 24, 35, 32, 45, 40, 55, 48, 65, 56, 75, 64, 85, 72, 95, 80, 105, 88, 115, 96, 125, 104, 135, 112, 145, 120, 155, 128, 165, 136, 175, 144, 185, 152, 195, 160, 205, 168, 215, 176, 225, 184, 235, 192, 245, 200, 255, 208

Offset: 0

Author

Paul Boddington, Jan 22 2004

Keywords

• easy
• nonn

Comments

a(n) is also the number of orbits of length n for T^2, if T is a map with n orbits of length n. - Thomas Ward, Apr 08 2009

Examples

			a(2) = (1/2)*mu(2)*sigma_2(2)+(1/2)*mu(1)*sigma_2(4) = 8. - _Thomas Ward_, Apr 08 2009
		

References

• I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS, May 1997, Vol. 44, Number 5.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4.
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)

Crossrefs

Cf. A091571, A091572, A091573, A091575, A091576, A091577.

Programs

Mathematica

CoefficientList[ Series[ (5 + 8x + 5x^2) / (1 - 2x^2 + x^4), {x, 0, 51}], x] (* Jean-François Alcover, Dec 02 2011 *)
With[{nn=40},Riffle[10*Range[nn]-5,8*Range[nn]]] (* or *) LinearRecurrence[ {0,2,0,-1},{5,8,15,16},80] (* Harvey P. Dale, Oct 30 2013 *)

PARI

(1/n)*sumdiv(n,d,moebius(n/d)*sumdiv(2*d,e,e^2)) \\ Thomas Ward, Apr 08 2009

Formula

a(n) = 5*(2*n+1) if n even, 4*(n+1) if n odd.

G.f.: (5+8*x+5*x^2)/(1-x^2)^2.

a(n) = (1/n)*Sum_{d|n} mobius(n/d)*sigma_2(2*d). - Thomas Ward, Apr 08 2009