This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A192355 #14 Sep 08 2022 08:45:57
%S A192355 1,0,6,2,70,90,926,2002,13110,37130,194446,640002,2973350,10653370,
%T A192355 46333566,174174002,730176790,2820264810,11582386286,45425564002,
%U A192355 184414199430,729520967450,2942491360606,11696742970002,47006639297270,187367554937290
%N A192355 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+2)^n+(x-2)^n) by x^2->x+2.
%C A192355 For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.
%C A192355 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 A192355 G. C. Greubel, <a href="/A192355/b192355.txt">Table of n, a(n) for n = 1..1000</a>
%H A192355 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,11,-12).
%F A192355 Empirical G.f.: x*(2*x^3-5*x^2-2*x+1)/((x-1)*(3*x+1)*(4*x-1)). - _Colin Barker_, Sep 12 2012
%F A192355 From _G. C. Greubel_, Oct 28 2018: (Start)
%F A192355 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k) * 4^k * phi_{n-2*k}, where phi_{n} = A078008(n).a(n) = (4^n + 2*(-3)^n + 2 + 2^n*delta(n,0))/6, with delta(n,0) = 1 if n=0, 0 else. (End)
%e A192355 (See A192352 for a related example.)
%t A192355 q[x_] := x + 2; d = 2;
%t A192355 p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)
%t A192355 Table[Expand[p[n, x]], {n, 0, 6}]
%t A192355 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192355 x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192355 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
%t A192355 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
%t A192355 (* A192355 *)
%t A192355 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
%t A192355 (* A192356 *)
%t A192355 Join[{1}, Table[(4^n + 2*(-3)^n + 2)/6, {n, 1, 50}]] (* _G. C. Greubel_, Oct 20 2018 *)
%o A192355 (PARI) for(n=0, 50, print1(if(n==0, 1, (4^n + 2*(-3)^n + 2)/6), ", ")) \\ _G. C. Greubel_, Oct 20 2018
%o A192355 (Magma) [1] cat [(4^n + 2*(-3)^n + 2)/6: n in [1..50]]; // _G. C. Greubel_, Oct 20 2018
%Y A192355 Cf. A192232, A192356.
%K A192355 nonn
%O A192355 1,3
%A A192355 _Clark Kimberling_, Jun 29 2011