A192375 -id:A192375 - cp's OEIS frontend

A192373 Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

1, 0, 7, 8, 77, 192, 1043, 3472, 15529, 57792, 240655, 934808, 3789653, 14963328, 60048443, 238578976, 953755537, 3798340224, 15162325975, 60438310184, 241126038941, 961476161856, 3835121918243, 15294304429744, 61000836720313, 243280700771904

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

The polynomials are given by p(n,x)=((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+4).

For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=4+x+3x^2 -> 7+4x
p(3,x)=16x+4x^2+4x^3 -> 8+28x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 77+84x.
From these, read A192352=(1,0,7,8,77,...) and A049602=(0,2,4,28,84,...).

Cf. A192232, A192374, A192375, A162517.

q[x_] := x + 1; d = Sqrt[x + 4];
p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d) (* A162517 *)
Table[Expand[p[n, x]], {n, 1, 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, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192373 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192374 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192375 *)

Conjecture: a(n) = 2*a(n-1)+10*a(n-2)-6*a(n-3)-9*a(n-4). G.f.: -x*(x+1)*(3*x-1) / (9*x^4+6*x^3-10*x^2-2*x+1). - Colin Barker, May 09 2014

A192374 Coefficient of x in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

Original entry on oeis.org

0, 2, 4, 28, 84, 406, 1448, 6200, 23688, 97034, 380716, 1533844, 6079452, 24339742, 96844496, 386805104, 1541301648, 6150529682, 24521644756, 97819530508, 390080615652, 1555871900710, 6204937972088, 24747735482792, 98698893741336

Offset: 1

Clark Kimberling, Jun 29 2011

nonn

The polynomials p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+4). A192374=2*A192375. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

			(See A192373.)

Cf. A192232, A192373, A192375.

Mathematica
```
(See A192373.)
```

Empirical G.f.: 2*x^2/(9*x^4+6*x^3-10*x^2-2*x+1). [Colin Barker, Nov 23 2012]

cp's OEIS Frontend

A192373 Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

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