A115951 → D

nonnegative coefficients

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

Recurrence

\(\left(2\,x - n - 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-6 - 18\,x - 12\,x^{2} + 8\,n + 14\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-4\,x - 12\,x^{2} - 8\,x^{3} + 4\,n x + 4\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	2	12													14
3	0	24	60												84
4	0	12	180	280											472
5	0	0	180	1120	1260										2560
6	0	0	60	1680	6300	5544									13584
7	0	0	0	1120	12600	33264	24024								71008
8	0	0	0	280	12600	83160	168168	102960							367168
9	0	0	0	0	6300	110880	504504	823680	437580						1882944
10	0	0	0	0	1260	83160	840840	2882880	3938220	1847560					9593920
11	0	0	0	0	0	33264	840840	5765760	15752880	18475600	7759752				48628096
12	0	0	0	0	0	5544	504504	7207200	36756720	83140200	85357272	32449872			245421312
13	0	0	0	0	0	0	168168	5765760	55135080	221707200	426786360	389398464	135207800		1234168832
14	0	0	0	0	0	0	24024	2882880	55135080	387987600	1280359080	2141691552	1757701400	561632400	6187414016

Technical details

Node id: n-fbfc644b975c7953 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-e60f1a95f04fff24

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: 2, 12;
  3: 0, 24,  60;
  4: 0, 12, 180,  280;
  5: 0,  0, 180, 1120,  1260;
  6: 0,  0,  60, 1680,  6300,   5544;
  7: 0,  0,   0, 1120, 12600,  33264,  24024;
  8: 0,  0,   0,  280, 12600,  83160, 168168, 102960;
  9: 0,  0,   0,    0,  6300, 110880, 504504, 823680, 437580;

%F Formula

The row polynomials satisfy (n^2-n-(2*n-2)*x)*P(n,x) = -(2*n^2-8*n+6-(14*n-18)*x+12*x^2)*P(n-1,x) + ((4*n-4)*x+(4*n-12)*x^2-8*x^3)*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[(-(2*n^2 - 8*n + 6 - (14*n - 18)*x + 12*x^2)*P[n - 1, x] + ((4*n - 4)*x + (4*n - 12)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(n^2 - n - (2*n - 2)*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 - 8*n + 6 - (14*n - 18)*x + 12*x^2)*v[i - 1] + ((4*n - 4)*x + (4*n - 12)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(n^2 - n - (2*n - 2)*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 Expansion of 1/sqrt(1-4*x*y-4*x^2*y).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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.