A039991 → D

Proved for all n: all roots real what this means

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(-3\,n + 2\,n x^{2} + 2\,n^{2} - n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-n x + n x^{3}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = -2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	P_n(1)
0							0
1							0
2	0	-2					-2
3	0	-6					-6
4	0	-16	0	4			-12
5	0	-40	0	20			-20
6	0	-96	0	72	0	-6	-30
7	0	-224	0	224	0	-42	-42

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-02ad05e8959bc17f · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-def41fdfbf381d0a

Observed on the stored rows ?

Degrees: -1, -1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13 (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: 0;
  1: 0;
  2: 0,    -2;
  3: 0,    -6;
  4: 0,   -16, 0,    4;
  5: 0,   -40, 0,   20;
  6: 0,   -96, 0,   72, 0,   -6;
  7: 0,  -224, 0,  224, 0,  -42;
  8: 0,  -512, 0,  640, 0, -192, 0,  8;
  9: 0, -1152, 0, 1728, 0, -720, 0, 72;

%F Formula

The row polynomials satisfy (n^2-3*n+2)*P(n,x) = (2*n^2-3*n-(n^2-2*n)*x^2)*P(n-1,x) - (n*x-n*x^3)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = -2*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := -2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 3*n - (n^2 - 2*n)*x^2)*P[n - 1, x] - (n*x - n*x^3)*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = -2*x; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^2 - 3*n - (n^2 - 2*n)*x^2)*v[i - 1] - (n*x - n*x^3)*deriv(v[i - 1], x))/(n^2 - 3*n + 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-6f899a70e4c27d14

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

preserver_image derivative_polynomial, shift 0 e-6f899a70e4c27d14

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of cos(x)^n in polynomial for cos(nx).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-9aecc6cc430e81f1 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

Same fingerprint (3)

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

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023