Real-rooted atlas

A095704: Triangle read by rows giving coefficients of the trigonometric expansion of sin(n*x).

A095704 ↗

Proved for all n all roots real

Recurrence from A095704

\(P_{n}(x) = \left(2\right)\,P_{n-1}(x) + \left(-1 - x^{2}\right)\,P_{n-2}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
111
222
330-12
440-40
550-1001-4
660-2006-8
770-350210-1-8
880-560560-80

All 15 rows b-files: A099087 (row sums) ?

Technical details
Node id
n-20136f1eca6c24ba · revision 0 · JSON
Definition source
oeis_catalog:A095704
Fingerprint
f6-52f747cbc56100f1

Observed on the stored rows ?

  • Degrees: 0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14 (irregular)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

Row sums ? A099087, A009545

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  2;
   3:  3, 0,   -1;
   4:  4, 0,   -4;
   5:  5, 0,  -10, 0,   1;
   6:  6, 0,  -20, 0,   6;
   7:  7, 0,  -35, 0,  21, 0,   -1;
   8:  8, 0,  -56, 0,  56, 0,   -8;
   9:  9, 0,  -84, 0, 126, 0,  -36, 0,  1;
  10: 10, 0, -120, 0, 252, 0, -120, 0, 10;
%F Formula
The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1+x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 + x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = 2*v[i - 1] - (1 + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

  • Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

  1. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003