A137338 → D

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

Recurrence

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

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

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

\(P_{2}(x) = -3 + 2\,x\)

\(P_{3}(x) = 6 - 12\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	-3	2						-1
3	6	-12	3					-3
4	-9	52	-30	4				17
5	3	-218	213	-60	5			-57
6	81	1002	-1425	620	-105	6		179
7	-786	-5326	9987	-5820	1475	-168	7	-631

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-68463e7ee4b23199 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-52366d9725e8b82c

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 -1)
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:      1;
  2:     -3,       2;
  3:      6,     -12,      3;
  4:     -9,      52,    -30,       4;
  5:      3,    -218,    213,     -60,      5;
  6:     81,    1002,  -1425,     620,   -105,      6;
  7:   -786,   -5326,   9987,   -5820,   1475,   -168,    7;
  8:   6711,   33020, -76485,   54916, -18235,   3066, -252,    8;
  9: -60309, -235824, 648012, -543844, 219485, -47964, 5782, -360, 9;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := -3 + 2*x;
P[3, x_] := 6 - 12*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 3*n + 2)*P[n - 4, x] - (2*n^2 - 5*n + 3 - (2*n - 2)*x)*P[n - 3, x] - (n^2 - (2*n - 2)*x + x^2)*P[n - 2, x] - (2*n - 1 - 2*x)*P[n - 1, x]];
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] = 1; v[3] = -3 + 2*x; v[4] = 6 - 12*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 3*n + 2)*v[i - 4] - (2*n^2 - 5*n + 3 - (2*n - 2)*x)*v[i - 3] - (n^2 - (2*n - 2)*x + x^2)*v[i - 2] - (2*n - 1 - 2*x)*v[i - 1]); 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-c9e97b9a66627f15
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-c9e97b9a66627f15

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-c9e97b9a66627f15

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), or…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-fe1ae6008805b5f0 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