A053122 → D

Proved for all n: all roots real · interlacing (weak) what this means

Recurrence

\(\left(-4 + 2\,n\right)\,P_{n}(x) = \left(-2 + x - 2\,n + n x\right)\,P_{n-1}(x) + \left(-4\,x + x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	-4	2													-2
4	10	-12	3												1
5	-20	42	-24	4											2
6	35	-112	108	-40	5										-4
7	-56	252	-360	220	-60	6									2
8	84	-504	990	-880	390	-84	7								3
9	-120	924	-2376	2860	-1820	630	-112	8							-6
10	165	-1584	5148	-8008	6825	-3360	952	-144	9						3
11	-220	2574	-10296	20020	-21840	14280	-5712	1368	-180	10					4
12	286	-4004	19305	-45760	61880	-51408	27132	-9120	1890	-220	11				-8
13	-364	6006	-34320	97240	-159120	162792	-108528	47880	-13860	2530	-264	12			4
14	455	-8736	58344	-194480	377910	-465120	379848	-210672	79695	-20240	3300	-312	13		5
15	-560	12376	-95472	369512	-839800	1220940	-1193808	807576	-382536	126500	-28600	4212	-364	14	-10

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Technical details

Node id: n-eaa24039c2a7a3ab · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-a89afae6aedee777

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:    1;
   3:   -4,     2;
   4:   10,   -12,     3;
   5:  -20,    42,   -24,     4;
   6:   35,  -112,   108,   -40,     5;
   7:  -56,   252,  -360,   220,   -60,     6;
   8:   84,  -504,   990,  -880,   390,   -84,    7;
   9: -120,   924, -2376,  2860, -1820,   630, -112,    8;
  10:  165, -1584,  5148, -8008,  6825, -3360,  952, -144, 9;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(2*n + 2 - (n + 1)*x)*v[i - 1] - (4*x - x^2)*deriv(v[i - 1], x))/(2*n - 4)); 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-b10ae33ac9ab7fa0
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-b10ae33ac9ab7fa0

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-b10ae33ac9ab7fa0

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing 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-46262d3eff45ae1d root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-8de75f6dc51d73d9 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