A039991: Triangle of coefficients of cos(x)^n in polynomial for cos(nx).

Proved for all n: all roots real what this means

Recurrence from OEIS A039991

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

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	2	0	-1					1
3	4	0	-3					1
4	8	0	-8	0	1			1
5	16	0	-20	0	5			1
6	32	0	-48	0	18	0	-1	1
7	64	0	-112	0	56	0	-7	1

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-dbf182f3ecfeda2a · revision 1 · JSON
Definition source: oeis_catalog:A039991
Fingerprint: f6-3d213e09756fa2d2

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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   1;
  1:   1;
  2:   2, 0,   -1;
  3:   4, 0,   -3;
  4:   8, 0,   -8, 0,   1;
  5:  16, 0,  -20, 0,   5;
  6:  32, 0,  -48, 0,  18, 0,   -1;
  7:  64, 0, -112, 0,  56, 0,   -7;
  8: 128, 0, -256, 0, 160, 0,  -32, 0, 1;
  9: 256, 0, -576, 0, 432, 0, -120, 0, 9;

%F Formula

The row polynomials satisfy P(n,x) = 2*P(n-1,x) - x^2*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - x^2*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 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 - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-9aecc6cc430e81f1

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

theorem_check three_term_recurrence e-9aecc6cc430e81f1 root: theorem check

Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Used in 3 proof steps

e-2abd4179002ed34f preserver_image (affine, shift 0) → shift_x_plus_one of Triangle of coefficients of cos(x)^n in polynomial for cos(nx).
e-6f899a70e4c27d14 preserver_image (derivative_polynomial, shift 0) → derivative of Triangle of coefficients of cos(x)^n in polynomial for cos(nx).
e-77a19b2535dc4780 preserver_image (derivative_polynomial, shift 0) → one_plus_derivative of Triangle of coefficients of cos(x)^n in polynomial for cos(nx).

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

chebyshev_t

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023