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A137234 Expansion of g.f. 1/((1-x)^2*(1 - 3*x + 2*x^2 - x^3)).

%I A137234 #26 Jun 07 2021 15:15:59
%S A137234 1,5,16,43,107,257,607,1422,3318,7727,17978,41810,97214,226014,525439,
%T A137234 1221519,2839710,6601549,15346765,35676927,82938821,192809396,
%U A137234 448227496,1042002541,2422362052,5631308596,13091204252,30433357644,70748973053
%N A137234 Expansion of g.f. 1/((1-x)^2*(1 - 3*x + 2*x^2 - x^3)).
%C A137234 Previous name: Transform of A000292 without the initial 0 by the T_{0,0} transformation (see link).
%C A137234 Partial sums of A137229. - _R. J. Mathar_, Nov 04 2008
%H A137234 G. C. Greubel, <a href="/A137234/b137234.txt">Table of n, a(n) for n = 0..1000</a>
%H A137234 Richard Choulet, <a href="https://www.apmep.fr/IMG/pdf/curtz1.pdf">Curtz-like transformation</a>.
%H A137234 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,8,-4,1).
%F A137234 O.g.f: 1/((1-z)^2*(1 - 3*z + 2*z^2 - z^3)).
%F A137234 a(n) = -(n+3) + Sum_{j=0..2} (-1)^j*(4-j)*A095263(n-j). - _G. C. Greubel_, Apr 19 2021
%t A137234 LinearRecurrence[{5,-9,8,-4,1}, {1,5,16,43,107}, 41] (* _G. C. Greubel_, Apr 19 2021 *)
%t A137234 CoefficientList[Series[1/((1-x)^2(1-3x+2x^2-x^3)),{x,0,30}],x] (* _Harvey P. Dale_, Jun 07 2021 *)
%o A137234 (Magma) I:=[1,5,16,43,107]; [n le 5 select I[n] else 5*Self(n-1) -9*Self(n-2) +8*Self(n-3) -4*Self(n-4) +Self(n-5): n in [1..41]]; // _G. C. Greubel_, Apr 19 2021
%o A137234 (Sage)
%o A137234 @CachedFunction
%o A137234 def A095263(n): return sum(binomial(n+j+2, 3*j+2) for j in (0..n//2))
%o A137234 def A137234(n): return -(n+3) + sum( (-1)^j*(4-j)*A095263(n-j) for j in (0..2))
%o A137234 [A137234(n) for n in (0..40)] # _G. C. Greubel_, Apr 19 2021
%Y A137234 Cf. A095263, A136302, A136303, A136304, A136305.
%K A137234 easy,nonn
%O A137234 0,2
%A A137234 _Richard Choulet_, Apr 05 2008