A142709 Polynomials made from cycloidal standing waves: p(x,n) = (4*n + 2)*x + x^n.

1, 2, 0, 7, 0, 10, 1, 0, 14, 0, 1, 0, 18, 0, 0, 1, 0, 22, 0, 0, 0, 1, 0, 26, 0, 0, 0, 0, 1, 0, 30, 0, 0, 0, 0, 0, 1, 0, 34, 0, 0, 0, 0, 0, 0, 1, 0, 38, 0, 0, 0, 0, 0, 0, 0, 1, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

1,2

Comments

Row sums are: {3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43};

The cycloid graphic parametrics are:

fc(t,n)=(2-1/n)*Cos[t]/2+Cos[(n-1)*t]/2*n;

fs(t,n)=(2-1/n)*Sin[t]/2+Sin[(n-1)*t]/2*n;

fe(t,n)=(2-1/n)*Exp[I*t]/2+Exp[I*(n-1)*t]/2*n;

Substitutions of x->Exp[i*x] and m->n-1 and multiplication by 2*n give the polynomials.

Links

Table of n, a(n) for n=1..67.

Formula

p(x,n) = (4*n + 2)*x + x^n; t(n,m) = Coefficients(p(x,n)).

Example

Triangle begins:
  {1, 2},
  {0, 7},
  {0, 10, 1},
  {0, 14, 0, 1},
  {0, 18, 0, 0, 1},
  {0, 22, 0, 0, 0, 1},
  {0, 26, 0, 0, 0, 0, 1},
  {0, 30, 0, 0, 0, 0, 0, 1},
  {0, 34, 0, 0, 0, 0, 0, 0, 1},
  {0, 38, 0, 0, 0, 0, 0, 0, 0, 1},
  {0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 1}

Mathematica

Clear[p, x, n, m] p[x_, n_] = (4*n + 2)*x + x^n; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%]

Crossrefs

Sequence in context: A285174 A269430 A363891 * A266220 A022897 A192496

Adjacent sequences: A142706 A142707 A142708 * A142710 A142711 A142712

Keyword

nonn,uned,tabf

Author

Roger L. Bagula, Sep 25 2008

Status

approved