chebyshev_t → x ↦ x+1 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	4	4													8
3	9	24	12												45
4	16	80	96	32											224
5	25	200	420	320	80										1045
6	36	420	1344	1728	960	192									4680
7	49	784	3528	6720	6160	2688	448								20377
8	64	1344	8064	21120	28160	19968	7168	1024							86912
9	81	2160	16632	57024	102960	104832	60480	18432	2304						364905
10	100	3300	31680	137280	320320	436800	358400	174080	46080	5120					1513160
11	121	4840	56628	302016	880880	1537536	1675520	1148928	481536	112640	11264				6211909
12	144	6864	96096	617760	2196480	4752384	6580224	5953536	3502080	1290240	270336	24576			25290720
13	169	9464	156156	1189760	5056480	13238784	22573824	25798656	19918080	10250240	3367936	638976	53248		102251773
14	196	12740	244608	2178176	10890880	33860736	69457920	97241088	94381056	63472640	29016064	8601600	1490944	114688	410963336

Technical details

Node id: n-45560eec68fca4d5 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d6d34c33242db792

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): yes
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  0;
  1:  1;
  2:  4,    4;
  3:  9,   24,    12;
  4: 16,   80,    96,    32;
  5: 25,  200,   420,   320,     80;
  6: 36,  420,  1344,  1728,    960,    192;
  7: 49,  784,  3528,  6720,   6160,   2688,   448;
  8: 64, 1344,  8064, 21120,  28160,  19968,  7168,  1024;
  9: 81, 2160, 16632, 57024, 102960, 104832, 60480, 18432, 2304;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n^2*x)*P[n - 1, x] + (2*n*x + n*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];
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; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + n^2*x)*v[i - 1] + (2*n*x + n*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from shift_x_plus_one of chebyshev_t?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from chebyshev_t?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

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