cp's OEIS Frontend

A199627 G.f.: (1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4)) for g=1.

%I A199627 #29 Sep 08 2022 08:46:00
%S A199627 1,2,1,1,2,2,4,7,8,9,12,15,16,17,20,23,24,25,28,31,32,33,36,39,40,41,
%T A199627 44,47,48,49,52,55,56,57,60,63,64,65,68,71,72,73,76,79,80,81,84,87,88,
%U A199627 89,92,95,96,97,100,103,104,105,108,111
%N A199627 G.f.: (1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4)) for g=1.
%C A199627 Expansion of a Poincaré series [or Poincare series] for space of moduli M_2 of stable bundles.
%H A199627 Georg Fischer, <a href="/A199627/b199627.txt">Table of n, a(n) for n = 0..999</a>
%H A199627 Bott, Raoul, <a href="http://dx.doi.org/10.1090/S0273-0979-1982-15038-8">Lectures on Morse theory, old and new</a>, Bull. Amer. Math. Soc. 7 (1982), no. 2, 331-358; reprinted in Vol. 48 (October, 2011). See Eq. (4.30).
%H A199627 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F A199627 a(n) = A047538(n-3) for n >= 6. - _Georg Fischer_, Oct 28 2018
%F A199627 From _Colin Barker_, Nov 05 2019: (Start)
%F A199627 G.f.: (1 + x)^2*(1 - 2*x + 2*x^2 - x^3 - x^4 + 3*x^5 - 2*x^6 + x^7) / ((1 - x)^2*(1 + x^2)).
%F A199627 a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>9.
%F A199627 a(n) = (-16 + (-i)^(1+n) + i^(1+n) + 4*n) / 2 for n>5, where i=sqrt(-1).
%F A199627 (End)
%p A199627 f:=g->(1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4));
%p A199627 s:=g->seriestolist(series(f(g),x,60));
%p A199627 s(1);
%o A199627 (Magma) g:=1; m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4))));  // _Bruno Berselli_, Nov 08 2011
%o A199627 (PARI) Vec((1 + x)^2*(1 - 2*x + 2*x^2 - x^3 - x^4 + 3*x^5 - 2*x^6 + x^7) / ((1 - x)^2*(1 + x^2)) + O(x^70)) \\ _Colin Barker_, Nov 05 2019
%Y A199627 Cf. A047538.
%K A199627 nonn,easy
%O A199627 0,2
%A A199627 _N. J. A. Sloane_, Nov 08 2011