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.

1, 0, 6, 2, 70, 90, 926, 2002, 13110, 37130, 194446, 640002, 2973350, 10653370, 46333566, 174174002, 730176790, 2820264810, 11582386286, 45425564002, 184414199430, 729520967450, 2942491360606, 11696742970002, 47006639297270, 187367554937290

Offset: 1

Author

Clark Kimberling, Jun 29 2011

Keywords

• nonn

Comments

For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

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

Examples

			(See A192352 for a related example.)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,11,-12).

Crossrefs

Cf. A192232, A192356.

Programs

Magma

[1] cat [(4^n + 2*(-3)^n + 2)/6: n in [1..50]]; // G. C. Greubel, Oct 20 2018

Mathematica

q[x_] := x + 2; d = 2;
p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
   x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
  (* A192355 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
  (* A192356 *)
Join[{1}, Table[(4^n + 2*(-3)^n + 2)/6, {n, 1, 50}]] (* G. C. Greubel, Oct 20 2018 *)

PARI

for(n=0, 50, print1(if(n==0, 1, (4^n + 2*(-3)^n + 2)/6), ", ")) \\ G. C. Greubel, Oct 20 2018

Formula

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

From G. C. Greubel, Oct 28 2018: (Start)

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)