A192376 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

1, 0, 7, 16, 73, 256, 975, 3616, 13521, 50432, 188247, 702512, 2621849, 9784832, 36517535, 136285248, 508623521, 1898208768, 7084211623, 26438637648, 98670339049, 368242718464, 1374300534895, 5128959421024, 19141537149297, 71437189176064

Offset

1,3

Comments

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

Links

Table of n, a(n) for n=1..26.

Formula

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)+2*a(n-3)-a(n-4). G.f.: x*(x-1)^2 / ((x+1)^2*(x^2-4*x+1)). - Colin Barker, May 11 2014

Example

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 -> 16+20x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 73+68x.
From these, read A192376=(1,0,7,16,73,...) and A192377=(0,2,4,20,68,...).

Mathematica

q[x_] := x + 2; d = Sqrt[x + 1];
p[n_, x_] := ((x + d)^n - (x - d)^ n )/(2 d)   (* Cf. 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}] (* A192376 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192377 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192378 *)

Crossrefs

Cf. A192232, A192377, A192378.

Sequence in context: A318481 A215180 A309476 * A363173 A214904 A029498

Adjacent sequences: A192373 A192374 A192375 * A192377 A192378 A192379

Keyword

nonn

Author

Clark Kimberling, Jun 29 2011

Status

approved