A053123 → D

Proved for all n: all roots real what this means

Recurrence

\(\left(6 - 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(3 - 12\,x - 5\,n + 19\,n x + 2\,n^{2} - 6\,n^{2} x\right)\,P_{n-1}(x) + \left(x - 4\,x^{2} - n x + 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	-2							-2
3	-4	6						2
4	-6	20	-12					2
5	-8	42	-60	20				-6
6	-10	72	-168	140	-30			4
7	-12	110	-360	504	-280	42		4
8	-14	156	-660	1320	-1260	504	-56	-10

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-97ab161dcdd2c9b4 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-2cff9b1fa72c8bfd

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:   0;
   2:  -2;
   3:  -4,   6;
   4:  -6,  20,   -12;
   5:  -8,  42,   -60,   20;
   6: -10,  72,  -168,  140,    -30;
   7: -12, 110,  -360,  504,   -280,    42;
   8: -14, 156,  -660, 1320,  -1260,   504,   -56;
   9: -16, 210, -1092, 2860,  -3960,  2772,  -840,   72;
  10: -18, 272, -1680, 5460, -10010, 10296, -5544, 1320, -90;

%F Formula

The row polynomials satisfy (2*n^2-7*n+6)*P(n,x) = (2*n^2-5*n+3-(6*n^2-19*n+12)*x)*P(n-1,x) - ((n-1)*x-(4*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = -2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := -2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 5*n + 3 - (6*n^2 - 19*n + 12)*x)*P[n - 1, x] - ((n - 1)*x - (4*n - 4)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 7*n + 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = -2; for(n = 3, N, my(i = n); v[i] = ((2*n^2 - 5*n + 3 - (6*n^2 - 19*n + 12)*x)*v[i - 1] - ((n - 1)*x - (4*n - 4)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 7*n + 6)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-e72fe32e447a59b1

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-e72fe32e447a59b1

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-b3aa6b8d6262c436 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

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