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A142586 Binomial transform of A014217.

%I A142586 #28 Apr 14 2021 06:29:28
%S A142586 1,2,5,14,39,107,290,779,2079,5522,14615,38579,101634,267347,702455,
%T A142586 1844114,4838079,12686507,33254210,87141659,228301839,598026002,
%U A142586 1566300455,4101923939,10741568514,28126975907,73647747815,192833044754,504884940879,1321888886747
%N A142586 Binomial transform of A014217.
%C A142586 The second term in the k-th iterated differences is 2, 3, 6, 10, 17, 28, 46, ... = A001610(k+1).
%H A142586 Colin Barker, <a href="/A142586/b142586.txt">Table of n, a(n) for n = 0..1000</a>
%H A142586 Nickolas Hein and Jia Huang, <a href="https://arxiv.org/abs/1807.04623">Variations of the Catalan numbers from some nonassociative binary operations</a>, arXiv:1807.04623 [math.CO], 2018.
%H A142586 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,2).
%F A142586 From _R. J. Mathar_, Sep 22 2008: (Start)
%F A142586 G.f.: (1 - 3*x + 2*x^2 + x^3)/((1-3*x+x^2)*(1-2*x)).
%F A142586 a(n) = A005248(n) - 2^(n-1), n>0. (End)
%F A142586 a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3); a(0)=1, a(1)=2, a(2)=5, a(3)=14. - _Harvey P. Dale_, Aug 08 2011
%F A142586 a(n) = (-2^(-1+n) + ((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n) for n > 0. - _Colin Barker_, Jun 05 2017
%p A142586 1,seq(combinat[fibonacci](2*n+1) +combinat[fibonacci](2*n-1) -2^(n-1), n = 1..30); # _G. C. Greubel_, Apr 13 2021
%t A142586 CoefficientList[Series[(1-3x+2x^2+x^3)/((1-3x+x^2)(1-2x)),{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{5,-7,2},{2,5,14},30]] (* _Harvey P. Dale_, Aug 08 2011 *)
%o A142586 (PARI) Vec((1-3*x+2*x^2+x^3)/((1-3*x+x^2)*(1-2*x)) + O(x^30)) \\ _Colin Barker_, Jun 05 2017
%o A142586 (Magma) [1] cat [Lucas(2*n) - 2^(n-1): n in [1..30]]; // _G. C. Greubel_, Apr 13 2021
%o A142586 (Sage) [1]+[lucas_number2(2*n,1,-1) -2^(n-1) for n in (1..30)] # _G. C. Greubel_, Apr 13 2021
%Y A142586 Cf. A001610, A005248.
%K A142586 nonn,easy
%O A142586 0,2
%A A142586 _Paul Curtz_, Sep 21 2008
%E A142586 Edited and extended by _R. J. Mathar_, Sep 22 2008