chebyshev_u → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	4	8	3													15
3	8	24	20	4												56
4	16	64	84	40	5											209
5	32	160	288	224	70	6										780
6	64	384	880	960	504	112	7									2911
7	128	896	2496	3520	2640	1008	168	8								10864
8	256	2048	6720	11648	11440	6336	1848	240	9							40545
9	512	4608	17408	35840	43680	32032	13728	3168	330	10						151316
10	1024	10240	43776	104448	152320	139776	80080	27456	5148	440	11					564719
11	2048	22528	107520	291840	496128	548352	396032	183040	51480	8008	572	12				2107560
12	4096	49152	259072	788480	1532160	1984512	1736448	1018368	388960	91520	12012	728	13			7865521
13	8192	106496	614400	2072576	4533760	6741504	6945792	4961280	2418624	777920	155584	17472	910	14		29354524
14	16384	229376	1437696	5324800	12953600	21762048	25842432	21829632	13023360	5374720	1478048	254592	24752	1120	15	109552575

Technical details

Node id: n-8595e145b88fcc5a · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-d6d34c33242db792

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:   1;
  1:   2,    2;
  2:   4,    8,     3;
  3:   8,   24,    20,     4;
  4:  16,   64,    84,    40,     5;
  5:  32,  160,   288,   224,    70,     6;
  6:  64,  384,   880,   960,   504,   112,     7;
  7: 128,  896,  2496,  3520,  2640,  1008,   168,    8;
  8: 256, 2048,  6720, 11648, 11440,  6336,  1848,  240,   9;
  9: 512, 4608, 17408, 35840, 43680, 32032, 13728, 3168, 330, 10;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + (3*n - 1)*x)*P[n - 1, x] - (x + 2*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + (3*n - 1)*x)*v[i - 1] - (x + 2*x^2)*deriv(v[i - 1], x))/(n)); 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from shift_x_plus_one of chebyshev_u?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from chebyshev_u?
    
    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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023