A053117 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	2							4
3	4	8	3						15
4	8	24	20	4					56
5	16	64	84	40	5				209
6	32	160	288	224	70	6			780
7	64	384	880	960	504	112	7		2911
8	128	896	2496	3520	2640	1008	168	8	10864

All 15 rows

Technical details

Node id: n-fbf0c4d004490261 · 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 -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 = 1, …, ascending powers of x

%e Example

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

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(3*n-4)*x)*P(n-1,x) - (x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (3*n - 4)*x)*P[n - 1, x] - (x + 2*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((2*n - 2 + (3*n - 4)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing or…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).?
    
    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