A136668 → D+1

Proved for all n all roots real interlacing (weak)

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(n^{2} - n^{3}\right)\,P_{n-4}(x) + \left(-8\,n^{2} x + 4\,n^{3} x\right)\,P_{n-3}(x) + \left(2 + n - 8\,n x^{2} - 2\,n^{2} + 12\,n^{2} x^{2} - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x)\)

\(P_{1}(x) = -2 + 4\,x + 2\,x^{2}\)

\(P_{2}(x) = -11 - 11\,x + 24\,x^{2} + 8\,x^{3}\)

\(P_{3}(x) = 8 - 148\,x - 74\,x^{2} + 192\,x^{3} + 48\,x^{4}\)

\(P_{4}(x) = 119 + 119\,x - 1896\,x^{2} - 632\,x^{3} + 1920\,x^{4} + 384\,x^{5}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	-2	4	2								4
2	-11	-11	24	8							10
3	8	-148	-74	192	48						26
4	119	119	-1896	-632	1920	384					14
5	-48	3268	1634	-26432	-6608	23040	3840				-1306
6	-1409	-1409	72096	24032	-409920	-81984	322560	46080			-29954
7	384	-65596	-32798	1557248	389312	-7070976	-1178496	5160960	645120		-594842
8	18825	18825	-2223168	-741056	34834240	6966848	-134894592	-19270656	92897280	10321920	-12071534

All 15 rows

Technical details

Node id: n-54032b7469d85693 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-1ecbfdc214b9f59d

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:      -2,       4,        2;
   2:     -11,     -11,       24,         8;
   3:       8,    -148,      -74,       192,          48;
   4:     119,     119,    -1896,      -632,        1920,        384;
   5:     -48,    3268,     1634,    -26432,       -6608,      23040,        3840;
   6:   -1409,   -1409,    72096,     24032,     -409920,     -81984,      322560,       46080;
   7:     384,  -65596,   -32798,   1557248,      389312,   -7070976,    -1178496,     5160960,       645120;
   8:   18825,   18825, -2223168,   -741056,    34834240,    6966848,  -134894592,   -19270656,     92897280,    10321920;
   9:   -3840, 1333660,   666830, -68928512,   -17232128,  823129344,   137188224, -2826584064,   -353323008,  1857945600,   185794560;
  10: -283875, -283875, 64464648,  21488216, -2106389440, -421277888, 20690191872,  2955741696, -64620011520, -7180001280, 40874803200, 3715891200;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(n^3-n^2)*P(n-4,x) + (4*n^3-8*n^2)*x*P(n-3,x) - (2*n^2-n-2+(4*n^3-12*n^2+8*n)*x^2)*P(n-2,x) + (4*n^2-8*n)*x*P(n-1,x) for n >= 5, with P(1,x) = -2 + 4*x + 2*x^2, P(2,x) = -11 - 11*x + 24*x^2 + 8*x^3, P(3,x) = 8 - 148*x - 74*x^2 + 192*x^3 + 48*x^4, P(4,x) = 119 + 119*x - 1896*x^2 - 632*x^3 + 1920*x^4 + 384*x^5.

%t Mathematica

P[1, x_] := -2 + 4*x + 2*x^2;
P[2, x_] := -11 - 11*x + 24*x^2 + 8*x^3;
P[3, x_] := 8 - 148*x - 74*x^2 + 192*x^3 + 48*x^4;
P[4, x_] := 119 + 119*x - 1896*x^2 - 632*x^3 + 1920*x^4 + 384*x^5;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2)*P[n - 4, x] + (4*n^3 - 8*n^2)*x*P[n - 3, x] - (2*n^2 - n - 2 + (4*n^3 - 12*n^2 + 8*n)*x^2)*P[n - 2, x] + (4*n^2 - 8*n)*x*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = -2 + 4*x + 2*x^2; v[2] = -11 - 11*x + 24*x^2 + 8*x^3; v[3] = 8 - 148*x - 74*x^2 + 192*x^3 + 48*x^4; v[4] = 119 + 119*x - 1896*x^2 - 632*x^3 + 1920*x^4 + 384*x^5; for(n = 5, N, my(i = n); v[i] = (-(n^3 - n^2)*v[i - 4] + (4*n^3 - 8*n^2)*x*v[i - 3] - (2*n^2 - n - 2 + (4*n^3 - 12*n^2 + 8*n)*x^2)*v[i - 2] + (4*n^2 - 8*n)*x*v[i - 1])/(n - 2)); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, +∞)
by Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2) w…?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[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