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A192382
Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.
6
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.
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 30 2011
STATUS
approved