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A137249 Expansion of g.f. z*(2-2*z+z^2+z^3)/((1+z)*(1-3*z+2*z^2-z^3)).

%I A137249 #27 Apr 19 2021 10:19:31
%S A137249 2,2,7,15,37,84,197,456,1062,2467,5737,13335,31002,72069,167542,
%T A137249 389486,905447,2104907,4893317,11375580,26445017,61477204,142917162,
%U A137249 332242091,772369157,1795540447,4174125122,9703663625,22558281082
%N A137249 Expansion of g.f. z*(2-2*z+z^2+z^3)/((1+z)*(1-3*z+2*z^2-z^3)).
%C A137249 Previous name was: Transform of A033999 by the T_{0,1} transformation (see link).
%H A137249 G. C. Greubel, <a href="/A137249/b137249.txt">Table of n, a(n) for n = 1..1000</a>
%H A137249 Richard Choulet, <a href="https://www.apmep.fr/IMG/pdf/curtz1.pdf">Curtz-like transformation</a>.
%H A137249 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-1,1).
%F A137249 O.g.f: z*(2 -2*z +z^2 +z^3)/( (1+z)*(1-3*z+2*z^2-z^3) ).
%F A137249 a(n+4) = 2*a(n+3) + a(n+2) - a(n+1) + a(n).
%F A137249 From _G. C. Greubel_, Apr 11 2021: (Start)
%F A137249 a(n) = (4*(-1)^n + 10*A095263(n) - 12*A095263(n-1) + 11*A095263(n-2))/7.
%F A137249 a(n) = (1/7)*( 4*(-1)^n + Sum_{j=0..floor(n/2)} ( 10*binomial(n+j+2, 3*j+2) - 12*binomial(n+j+1, 3*j+2) + 11*binomial(n+j, 3*j+2) ) ). (End)
%p A137249 m:= 40;
%p A137249 S:= series( x*(2-2*x+x^2+x^3)/((1+x)*(1-3*x+2*x^2-x^3)), x, m+1);
%p A137249 seq(coeff(S, x, j), j = 1..m); # _G. C. Greubel_, Apr 11 2021
%t A137249 LinearRecurrence[{2,1,-1,1},{2,2,7,15},30] (* _Harvey P. Dale_, Feb 02 2012 *)
%o A137249 (Magma)
%o A137249 R<x>:=PowerSeriesRing(Integers(), 40);
%o A137249 Coefficients(R!( (2-2*x+x^2+x^3)/((1+x)*(1-3*x+2*x^2-x^3)) )); // _G. C. Greubel_, Apr 11 2021
%o A137249 (Sage)
%o A137249 def A132749_list(prec):
%o A137249     P.<x> = PowerSeriesRing(ZZ, prec)
%o A137249     return P( (2-2*x+x^2+x^3)/((1+x)*(1-3*x+2*x^2-x^3)) ).list()
%o A137249 A132749_list(40) # _G. C. Greubel_, Apr 11 2021
%Y A137249 Cf. A095263, A136302, A136303, A136304, A136305.
%K A137249 easy,nonn
%O A137249 1,1
%A A137249 _Richard Choulet_, Apr 05 2008
%E A137249 New name using g.f. from _Joerg Arndt_, Apr 19 2021