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A091574 Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type D_4. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

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)

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)*sumdiv(n,d,mobius(n/d)*sigma_2(2*d). - Thomas Ward, Apr 08 2009

EXAMPLE

a(2) = (1/2)mu(2)sigma_2(2)+(1/2)mu(1)sigma_2(4) = 8. - Thomas Ward, Apr 08 2009

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A112269 A314527 A314528 * A314529 A314530 A314531

Adjacent sequences:  A091571 A091572 A091573 * A091575 A091576 A091577

KEYWORD

easy,nonn

AUTHOR

Paul Boddington, Jan 22 2004

STATUS

approved

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Last modified August 3 19:49 EDT 2021. Contains 346441 sequences. (Running on oeis4.)