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A136303 Expansion of g.f. (1 +x^2)/((1-x)^2*(1 -3*x +2*x^2 -x^3)).

%I A136303 #15 Sep 08 2022 08:45:32
%S A136303 1,5,17,48,123,300,714,1679,3925,9149,21296,49537,115192,267824,
%T A136303 622653,1447533,3365149,7823068,18186475,42278476,98285586,228486323,
%U A136303 531166317,1234811937,2870589548,6673311137,15513566304,36064666240,83840177305
%N A136303 Expansion of g.f. (1 +x^2)/((1-x)^2*(1 -3*x +2*x^2 -x^3)).
%C A136303 Previous name: Transform of 0 by the reciprocal transformation to T_{1,1} (see link).
%H A136303 G. C. Greubel, <a href="/A136303/b136303.txt">Table of n, a(n) for n = 0..1000</a>
%H A136303 Richard Choulet, <a href="https://www.apmep.fr/IMG/pdf/curtz1.pdf">Curtz-like transformation</a>.
%H A136303 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,8,-4,1).
%F A136303 G.f.: f(z) = 1 +5*z + ... = (1+z^2)/((1-z)^2*(1-3*z+2*z^2-z^3)).
%F A136303 a(n+5) = 5*a(n+4) -9*a(n+3) +8*a(n+2) -4*a(n+1) +a(n) (n>=0). - _Richard Choulet_, Apr 07 2009
%F A136303 From _G. C. Greubel_, Apr 19 2021: (Start)
%F A136303 a(n) = -2*(n+2) + 5*A095263(n) - 4*A095263(n-1) + 2*A095263(n-2).
%F A136303 a(n) = -2*(n+2) + Sum_{k=0..floor(n/2)} (5*binomial(n+k+2, 3*k+2) - 4*binomial(n +k+1, 3*k+2) + 2*binomial(n+k, 3*k+2)). (End)
%p A136303 A136303:= n-> -2*(n+2) + add( (5*binomial(n+k+2, 3*k+2) - 4*binomial(n +k+1, 3*k+2) + 2*binomial(n+k, 3*k+2)), k=0..n/2 );
%p A136303 seq(A136303(n), n=0..40); # _G. C. Greubel_, Apr 19 2021
%t A136303 LinearRecurrence[{5,-9,8,-4,1},{1,5,17,48,123},40] (* _Harvey P. Dale_, Apr 01 2018 *)
%o A136303 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x^2)/((1-x)^2*(1-3*x+2*x^2-x^3)) )); // _G. C. Greubel_, Apr 19 2021
%o A136303 (Sage)
%o A136303 def A136303_list(prec):
%o A136303     P.<x> = PowerSeriesRing(ZZ, prec)
%o A136303     return P( (1+x^2)/((1-x)^2*(1-3*x+2*x^2-x^3)) ).list()
%o A136303 A136303_list(40) # _G. C. Greubel_, Apr 19 2021
%Y A136303 Cf. A095263, A097550, A135364, A136302, A136304, A136305, A137229, A137234, A137249.
%K A136303 nonn,easy
%O A136303 0,2
%A A136303 _Richard Choulet_, Mar 22 2008