A147310 A golden mean based polynomials set that behaves like an even powered Pascal triangle: p(x,n) = (x - phi)^floor(n/2)*(x + phi)^floor(n/2).

1, 1, 1, 0, -1, 1, 0, -1, 1, 0, -2, 0, 1, 1, 0, -2, 0, 1, 1, 0, -3, 0, 3, 0, -1, 1, 0, -3, 0, 3, 0, -1, 1, 0, -4, 0, 6, 0, -4, 0, 1, 1, 0, -4, 0, 6, 0, -4, 0, 1, 1, 0, -5, 0, 10, 0, -10, 0, 5, 0, -1

Offset

0,11

Links

Table of n, a(n) for n=0..60.

Formula

p(x,n)=(x - GoldenRatio)^Floor[n/2]*(x + GoldenRatio)^Floor[n/2]; t(n,m)=GoldenRatio^((-m + 1))*Reverse(Coefficients(p(x,n))).

Example

{1}, {1}, {1, 0, -1}, {1, 0, -1}, {1, 0, -2, 0, 1}, {1, 0, -2, 0, 1}, {1, 0, -3, 0, 3, 0, -1}, {1, 0, -3, 0, 3, 0, -1}, {1, 0, -4, 0, 6, 0, -4, 0, 1}, {1, 0, -4, 0, 6, 0, -4, 0, 1}, {1, 0, -5, 0, 10, 0, -10, 0, 5, 0, -1}

Mathematica

Clear[p, x, n]; p[x_, n_] = (x - GoldenRatio)^Floor[n/2]*(x + GoldenRatio)^Floor[n/2] a = Table[Reverse[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]], {n, 0, 10}]; Flatten[%]; b = Table[a[[n]][[m]]*GoldenRatio^((-m + 1)), {n, 1, Length[a]}, {m, 1, Length[a[[n]]]}]; Flatten[b]

Crossrefs

Sequence in context: A281271 A284319 A281272 * A025886 A117355 A319571

Adjacent sequences: A147307 A147308 A147309 * A147311 A147312 A147313

Keyword

tabf,sign,more,uned

Author

Roger L. Bagula, Nov 05 2008

Status

approved