A192382 Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.

0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, 1398784, 5591040, 22372352, 89473024, 357924864, 1431633920, 5726666752, 22906404864, 91626143744, 366503526400, 1466016202752, 5864060616704, 23456250855424, 93824986644480

Offset

1,2

Comments

The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+2). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+2, see A192232.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,8).

Formula

Conjectures from Colin Barker, May 12 2014: (Start)

a(n) = 2^(n-2)*(2*(-1)^n + 2^n)/3 = 2*A003683(n-1).

a(n) = 2*a(n-1) + 8*a(n-2).

G.f.: 2*x^2 / ((1+2*x)*(1-4*x)). (End).

a(n) = 4^n*(1 - (-1/2)^n)/3. - Peter Luschny, Oct 02 2019

E.g.f: (1/3)*(2 + exp(2*x))*(sinh(x))^2. - G. C. Greubel, Feb 19 2023

Example

The first five polynomials p(n,x) and their reductions are as follows:
  p(0, x) = 1 -> 1.
  p(1, x) = 2*x -> 2*x.
  p(2, x) = 2 + x + 3*x^2 -> 8 + 4*x.
  p(3, x) = 8*x + 4*x^2 + 4*x^3 -> 16 + 24*x.
  p(4, x) = 4 + 4*x + 21*x^2 + 10*x^3 + 5*x^4 -> 96 + 80*x.
From these, read A083086 = (1, 0, 9, 16, 96, ...) and A192382 =(0, 2, 4, 24, 80, ...).

Maple

seq(4^n*(1-(-1/2)^n)/3, n=0..24); # Peter Luschny, Oct 02 2019

Mathematica

q[x_]:= x+2; d= Sqrt[x+2];
p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2 d); (* suggested by A162517 *)
Table[Expand[p[n, x]], {n, 6}]
reductionRules= {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x*q[x]^((y- 1)/2)};
t = Table[FixedPoint[Expand[#1/. reductionRules] &, p[n, x]], {n, 30}];
Table[Coefficient[Part[t, n], x, 0], {n, 30}] (* abs value of A083086 *)
Table[Coefficient[Part[t, n], x, 1], {n, 30}] (* 2*A003683 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 30}] (* A003683 *)
LinearRecurrence[{2, 8}, {0, 2}, 40] (* G. C. Greubel, Feb 19 2023 *)

Prog

(Magma) [(4^(n-1) - (-2)^(n-1))/3: n in [1..40]]; // G. C. Greubel, Feb 19 2023
(SageMath) [(4^(n-1) - (-2)^(n-1))/3 for n in range(1, 41)] # G. C. Greubel, Feb 19 2023

Crossrefs

Cf. A003683, A083086, A162517, A192232.

Sequence in context: A192513 A192384 A119036 * A232205 A170931 A317999

Adjacent sequences: A192379 A192380 A192381 * A192383 A192384 A192385

Keyword

nonn

Author

Clark Kimberling, Jun 30 2011

Status

approved