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

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 ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	2															2
3	3	0	-1													2
4	4	0	-4													0
5	5	0	-10	0	1											-4
6	6	0	-20	0	6											-8
7	7	0	-35	0	21	0	-1									-8
8	8	0	-56	0	56	0	-8									0
9	9	0	-84	0	126	0	-36	0	1							16
10	10	0	-120	0	252	0	-120	0	10							32
11	11	0	-165	0	462	0	-330	0	55	0	-1					32
12	12	0	-220	0	792	0	-792	0	220	0	-12					0
13	13	0	-286	0	1287	0	-1716	0	715	0	-78	0	1			-64
14	14	0	-364	0	2002	0	-3432	0	2002	0	-364	0	14			-128
15	15	0	-455	0	3003	0	-6435	0	5005	0	-1365	0	105	0	-1	-128

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

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, +∞)
by Liu–Wang theorem for three-term recurrences

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

[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