A119808 → D

nonnegative coefficients

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

Recurrence

\(\left(5 - 2\,x - 5\,n - 4\,n x + 6\,n^{2} x\right)\,P_{n}(x) = \left(-7 + 14\,x - 16\,x^{2} - 9\,n + 8\,n x - 8\,n x^{2} + 10\,n^{2} x + 8\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-1 - 5\,x + 22\,x^{2} - 16\,x^{3} + 2\,n x + 14\,n x^{2} - 16\,n x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	5							5
3	17	4						21
4	49	32						81
5	129	156	12					297
6	321	600	132					1053
7	769	2004	840	32				3645
8	1793	6096	4056	448				12393
9	4097	17356	16500	3520	80			41553
10	9217	47048	59676	20480	1360			137781
11	20481	122820	198096	98400	12720	192		452709
12	45057	311232	616080	413376	86640	3840		1476225
13	98305	769980	1820508	1571584	480480	41664	448	4782969
14	212993	1867704	5163060	5530880	2301600	325248	10304	15411789

Technical details

Node id: n-a9089d8e5df0a6be · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-40f5e5ec9eda3abc

Observed on the stored rows ?

Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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:    5;
  3:   17,     4;
  4:   49,    32;
  5:  129,   156,    12;
  6:  321,   600,   132;
  7:  769,  2004,   840,   32;
  8: 1793,  6096,  4056,  448;
  9: 4097, 17356, 16500, 3520, 80;

%F Formula

The row polynomials satisfy -(5*n-5-(6*n^2-4*n-2)*x)*P(n,x) = -(9*n+7-(10*n^2+8*n+14)*x-(8*n^2-8*n-16)*x^2)*P(n-1,x) - (1-(2*n-5)*x-(14*n+22)*x^2+(16*n+16)*x^3)*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[Together[(-(9*n + 7 - (10*n^2 + 8*n + 14)*x - (8*n^2 - 8*n - 16)*x^2)*P[n - 1, x] - (1 - (2*n - 5)*x - (14*n + 22)*x^2 + (16*n + 16)*x^3)*D[P[n - 1, x], x])/(-5*n + 5 + (6*n^2 - 4*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(9*n + 7 - (10*n^2 + 8*n + 14)*x - (8*n^2 - 8*n - 16)*x^2)*v[i - 1] - (1 - (2*n - 5)*x - (14*n + 22)*x^2 + (16*n + 16)*x^3)*deriv(v[i - 1], x))/(-5*n + 5 + (6*n^2 - 4*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 Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceilin…?
  - 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.