A185410 → reverse → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-5\,x^{2} + 12\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2\,x^{2} - 2\,x^{3} - 4\,n x^{2} + 4\,n x^{3}\right)\,P_{n-2}'(x) + \left(2 - 6\,x + 4\,n x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	2	2													4
3	4	20	3												27
4	8	120	108	4											240
5	16	592	1548	464	5										2625
6	32	2656	15504	14032	1790	6									34020
7	64	11328	128880	256960	105600	6516	7								509355
8	128	46976	960192	3602240	3304400	711792	22904	8							8648640
9	256	191744	6677184	42900736	73569200	36298464	4472776	78656	9						164189025
10	512	775680	44377344	458943744	1325577120	1230128928	359056320	26817216	265626	10					3445942500
11	1024	3122176	286043904	4559096832	20713607040	32259165120	17943142224	3301526016	155628108	885620	11				79222218075
12	2048	12531712	1805245440	42956436480	292619136000	710242251264	665196610176	237382450560	28800823320	882313000	2922788	12			1979900722800
13	4096	50221056	11226221568	389384376320	3839015635200	13790051463168	20084760019968	12160590879744	2921912131680	241652663600	4917022836	9565776	13		53443570205025
14	8192	201089024	69089955840	3429409214464	47638035289600	243796536516096	522512540884992	492971602944000	203081730507072	34052174588800	1968900012112	27049703184	31089110	14	1549547301802500

Technical details

Node id: n-36edb24e07155eca · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-2fbd00207b225162

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:   2,      2;
  3:   4,     20,       3;
  4:   8,    120,     108,        4;
  5:  16,    592,    1548,      464,        5;
  6:  32,   2656,   15504,    14032,     1790,        6;
  7:  64,  11328,  128880,   256960,   105600,     6516,       7;
  8: 128,  46976,  960192,  3602240,  3304400,   711792,   22904,     8;
  9: 256, 191744, 6677184, 42900736, 73569200, 36298464, 4472776, 78656, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^2-12*n+5)*x^2*P(n-2,x) - ((4*n-2)*x^2-(4*n-2)*x^3)*d/dx P(n-2,x) + (2+(4*n-6)*x)*P(n-1,x) + (2*x-2*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[-(4*n^2 - 12*n + 5)*x^2*P[n - 2, x] - ((4*n - 2)*x^2 - (4*n - 2)*x^3)*D[P[n - 2, x], x] + (2 + (4*n - 6)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];
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] = -(4*n^2 - 12*n + 5)*x^2*v[i - 2] - ((4*n - 2)*x^2 - (4*n - 2)*x^3)*deriv(v[i - 2], x) + (2 + (4*n - 6)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); 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 A triangular decomposition of the double factorial numbers A001147.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A triangular decomposition of the double factorial numbers A001147.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from A decomposition of the double factorials A001147.?
    
    ↑ Liu–Wang theorem (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.