chebyshev_t → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	2	4	1													7
3	4	12	9	1												26
4	8	32	40	16	1											97
5	16	80	140	100	25	1										362
6	32	192	432	448	210	36	1									1351
7	64	448	1232	1680	1176	392	49	1								5042
8	128	1024	3328	5632	5280	2688	672	64	1							18817
9	256	2304	8640	17472	20592	14256	5544	1080	81	1						70226
10	512	5120	21760	51200	72800	64064	34320	10560	1650	100	1					262087
11	1024	11264	53504	143616	239360	256256	176176	75504	18876	2420	121	1				978122
12	2048	24576	129024	389120	744192	940032	792064	439296	154440	32032	3432	144	1			3650401
13	4096	53248	306176	1025024	2213120	3224832	3224832	2206464	1011296	297440	52052	4732	169	1		13623482
14	8192	114688	716800	2637824	6347264	10486784	12155136	9922560	5643456	2178176	544544	81536	6370	196	1	50843527

Technical details

Node id: n-cdf01fb0e5c8c5bd · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-d44a70ca53ce7ac3

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:   1,    1;
  2:   2,    4,    1;
  3:   4,   12,    9,     1;
  4:   8,   32,   40,    16,     1;
  5:  16,   80,  140,   100,    25,     1;
  6:  32,  192,  432,   448,   210,    36,    1;
  7:  64,  448, 1232,  1680,  1176,   392,   49,    1;
  8: 128, 1024, 3328,  5632,  5280,  2688,  672,   64,  1;
  9: 256, 2304, 8640, 17472, 20592, 14256, 5544, 1080, 81, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (3*n - 3)*x)*v[i - 1] - (x + 2*x^2)*deriv(v[i - 1], x))/(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 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_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.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023