A163936 → reverse → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,x^{2} + 10\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(x^{2} - x^{3} - 2\,n x^{2} + 2\,n x^{3}\right)\,P_{n-2}'(x) + \left(1 - 4\,x + 4\,n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	1	4													5
3	1	16	18												35
4	1	44	174	96											315
5	1	104	984	1776	600										3465
6	1	228	4356	17600	18540	4320									45045
7	1	480	16830	128480	290700	203904	35280								675675
8	1	988	59850	783200	3220100	4711824	2387952	322560							11486475
9	1	2008	201780	4250000	28827500	74640384	77184072	29871360	3265920						218243025
10	1	4052	656544	21304640	223825000	932144304	1672334328	1297495296	399051360	36288000					4583103525
11	1	8144	2085114	100975616	1571848600	9890305824	28018865616	37137311616	22567330944	5683564800	439084800				105411381075
12	1	16332	6511878	459505504	10257413880	93293764704	392289791488	809443464192	829798112016	407883018240	86104909440	5748019200			2635284526875
13	1	32712	20099088	2029037104	63349088880	805940521344	4814040324688	14545509138432	22948789260816	18830796618240	7677223804800	1384033029120	80951270400		71152682225625
14	1	65476	61523964	8759319232	375080261140	6508059074496	53443825045744	226399280528896	517449361181328	647283751398720	436585358613888	150610848215040	23542903238400	1220496076800	2063427784543125

Technical details

Node id: n-00fabd7673b77000 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-fbfce332e6b0be6b

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: 1,    4;
  3: 1,   16,     18;
  4: 1,   44,    174,      96;
  5: 1,  104,    984,    1776,      600;
  6: 1,  228,   4356,   17600,    18540,     4320;
  7: 1,  480,  16830,  128480,   290700,   203904,    35280;
  8: 1,  988,  59850,  783200,  3220100,  4711824,  2387952,   322560;
  9: 1, 2008, 201780, 4250000, 28827500, 74640384, 77184072, 29871360, 3265920;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^2-10*n+4)*x^2*P(n-2,x) - ((2*n-1)*x^2-(2*n-1)*x^3)*d/dx P(n-2,x) + (1+(4*n-4)*x)*P(n-1,x) + (x-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 - 10*n + 4)*x^2*P[n - 2, x] - ((2*n - 1)*x^2 - (2*n - 1)*x^3)*D[P[n - 2, x], x] + (1 + (4*n - 4)*x)*P[n - 1, x] + (x - 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 - 10*n + 4)*x^2*v[i - 2] - ((2*n - 1)*x^2 - (2*n - 1)*x^3)*deriv(v[i - 2], x) + (1 + (4*n - 4)*x)*v[i - 1] + (x - 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 E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).?
    
    ↑ Liu–Wang theorem (above)
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

Related ?

Same fingerprint ?

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.