chebyshev_u → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	2							2
2	8	8						16
3	20	48	24					92
4	40	168	192	64				464
5	70	448	864	640	160			2182
6	112	1008	2880	3520	1920	384		9824
7	168	2016	7920	14080	12480	5376	896	42936

All 15 rows

Technical details

Node id: n-d9a5325f33547000 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-5a38d53f8ebfe52d

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:   2;
  2:   8,    8;
  3:  20,   48,    24;
  4:  40,  168,   192,     64;
  5:  70,  448,   864,    640,    160;
  6: 112, 1008,  2880,   3520,   1920,    384;
  7: 168, 2016,  7920,  14080,  12480,   5376,    896;
  8: 240, 3696, 19008,  45760,  58240,  40320,  14336,  2048;
  9: 330, 6336, 41184, 128128, 218400, 215040, 121856, 36864, 4608;

%F Formula

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

%t Mathematica

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

A053117 → x ↦ x+1 → D

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