A137408 Triangular sequence from coefficients of a switched even -odd polynomial recursion: odd:p(x,n)=2*x*p(x, n - 1) - p(x, n - 2); even:p(x,n)=(1 - 2*x)*p(x, n - 1) - p(x, n - 2);.

1, 0, 2, -1, 2, -4, 0, -4, 4, -8, 1, -6, 16, -16, 16, 0, 6, -16, 40, -32, 32, -1, 12, -44, 88, -128, 96, -64, 0, -8, 40, -128, 208, -288, 192, -128, 1, -20, 100, -296, 592, -800, 832, -512, 256, 0, 10, -80, 328, -800, 1472, -1792, 1792, -1024, 512, -1

Offset

1,3

Comments

A048788 gives the row sums: {1, 2, -3, -8, 11, 30, -41, -112, 153, 418, -571}

Links

Table of n, a(n) for n=1..56.

Formula

p(x,-1)=0;p(x,0)=1;p(x,1]=2*x; odd:p(x,n)=2*x*p(x, n - 1) - p(x, n - 2); even:p(x,n)=(1 - 2*x)*p(x, n - 1) - p(x, n - 2);

Example

{1},
{0, 2},
{-1, 2, -4},
{0, -4, 4, -8},
{1, -6, 16, -16,16},
{0, 6, -16, 40, -32, 32},
{-1, 12, -44, 88, -128, 96, -64},
{0, -8, 40, -128, 208, -288, 192, -128},
{1, -20, 100, -296, 592, -800, 832, -512, 256},
{0,10, -80, 328, -800,1472, -1792, 1792, -1024, 512},
{-1, 30, -200, 784, -2048, 3872, -5568, 5888, -4864, 2560, -1024}

Mathematica

Clear[p, x, a] p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = 2*x; p[x_, n_] := p[x, n] = If[Mod[n, 2] == 1, 2*x*p[x, n - 1] - p[x, n - 2], (1 - 2*x)*p[x, n - 1] - p[x, n - 2]]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]

Crossrefs

Cf. A048788.

Sequence in context: A227428 A265255 A131022 * A007461 A181302 A304785

Adjacent sequences: A137405 A137406 A137407 * A137409 A137410 A137411

Keyword

tabl,uned,sign

Author

Roger L. Bagula, Apr 14 2008

Status

approved