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%I #39 Aug 25 2024 20:22:45
%S 5,8,15,16,25,24,35,32,45,40,55,48,65,56,75,64,85,72,95,80,105,88,115,
%T 96,125,104,135,112,145,120,155,128,165,136,175,144,185,152,195,160,
%U 205,168,215,176,225,184,235,192,245,200,255,208
%N Poincaré series [or Poincare series] of the preprojective algebra of an extended Dynkin diagram of type D_4.
%C 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
%D I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS, May 1997, Vol. 44, Number 5.
%H Harvey P. Dale, <a href="/A091574/b091574.txt">Table of n, a(n) for n = 0..1000</a>
%H Apisit Pakapongpun and Thomas Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Ward/ward17.html">Functorial orbit counting</a>, Journal of Integer Sequences, 12 (2009) Article 09.2.4.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1)
%F a(n) = 5*(2*n+1) if n even, 4*(n+1) if n odd.
%F G.f.: (5+8*x+5*x^2)/(1-x^2)^2.
%F a(n) = (1/n)*Sum_{d|n} mobius(n/d)*sigma_2(2*d). - _Thomas Ward_, Apr 08 2009
%e a(2) = (1/2)*mu(2)*sigma_2(2)+(1/2)*mu(1)*sigma_2(4) = 8. - _Thomas Ward_, Apr 08 2009
%t CoefficientList[ Series[ (5 + 8x + 5x^2) / (1 - 2x^2 + x^4), {x, 0, 51}], x] (* _Jean-François Alcover_, Dec 02 2011 *)
%t 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 *)
%o (PARI) (1/n)*sumdiv(n,d,moebius(n/d)*sumdiv(2*d,e,e^2)) \\ _Thomas Ward_, Apr 08 2009
%Y Cf. A091571, A091572, A091573, A091575, A091576, A091577.
%K easy,nonn
%O 0,1
%A _Paul Boddington_, Jan 22 2004