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A192356 Coefficients of x in the reduction of the polynomial p(n,x) = ((x+2)^n + (x-2)^n)/2 by x^2->x+2.

%I A192356 #36 Sep 08 2022 08:45:57
%S A192356 0,1,1,15,29,211,561,3095,9829,46971,164921,728575,2707629,11450531,
%T A192356 43942081,181348455,708653429,2884834891,11388676041,46006694735,
%U A192356 182670807229,734751144051,2926800830801,11743814559415,46865424529029,187791199242011,750176293590361
%N A192356 Coefficients of x in the reduction of the polynomial p(n,x) = ((x+2)^n + (x-2)^n)/2 by x^2->x+2.
%C A192356 For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.
%C A192356 Direct sums can be obtained for A192355 and A192356 in the following way. The polynomials p_{n}(x) can be given in series form by p_{n}(x) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*4*k*x^(n-2*k). For the reduction x^2 -> x+2 then the general form can be seen as x^n -> J_{n}*x + phi_{n}, where J_{n} = A001045(n) are the Jacobsthal numbers and phi_{n} = A078008. The reduction of p_{n}(x) now takes the form p_{n}(x) = x * Sum_{k=0..floor(n/2)} binomial(n,2*k)*4^k*J_{n-2*k} + Sum_{k=0..floor(n/2)} binomial(n,2*k)*4^k*phi_{n-2*k}. Evaluating the series leads to p_{n}(x) = x * (4^n - (-3)^n - 1 + 2^n*delta(n,0))/6 + (4^n + 2*(-3)^n + 2 + 2^n*delta(n,0))/6, where delta(n,k) is the Kronecker delta. - _G. C. Greubel_, Oct 29 2018
%H A192356 G. C. Greubel, <a href="/A192356/b192356.txt">Table of n, a(n) for n = 1..1000</a>
%H A192356 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,11,-12).
%F A192356 Empirical g.f.: x^2*(1-x+2*x^2)/((x-1)*(3*x+1)*(4*x-1)). - _Colin Barker_, Sep 12 2012
%F A192356 From _G. C. Greubel_, Oct 28 2018: (Start)
%F A192356 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * 4^k * J_{n-2*k}, where J_{n} = A001045(n) are the Jacobsthal numbers.
%F A192356 a(n) = (4^n - (-3)^n - 1 + 2^n*delta(n,0))/6, with delta(n,0) = 1 if n=0, 0 otherwise. (End)
%t A192356 (See A192355.)
%t A192356 Join[{0}, Table[(4^n - (-3)^n - 1)/6, {n, 1, 50}]] (* _G. C. Greubel_, Oct 20 2018 *)
%o A192356 (PARI) for(n=0, 50, print1(if(n==0, 0, (4^n - (-3)^n - 1)/6), ", ")) \\ _G. C. Greubel_, Oct 20 2018
%o A192356 (Magma) [0] cat [(4^n - (-3)^n - 1)/6: n in [1..50]]; // _G. C. Greubel_, Oct 20 2018
%Y A192356 Cf. A192232, A192355.
%K A192356 nonn
%O A192356 1,4
%A A192356 _Clark Kimberling_, Jun 29 2011