A206306 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\right)\,P_{n-4}(x) + \left(12 + 4\,x\right)\,P_{n-3}(x) + \left(-13 - 6\,x - x^{2}\right)\,P_{n-2}(x) + \left(6 + 2\,x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 3 + 2\,x\)

\(P_{3}(x) = 7 + 12\,x + 3\,x^{2}\)

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	3	2													5
3	7	12	3												22
4	15	46	27	4											92
5	31	144	144	48	5										372
6	63	402	594	328	75	6									1468
7	127	1044	2097	1680	625	108	7								5688
8	255	2582	6669	7180	3825	1062	147	8							21728
9	511	6168	19686	27072	19200	7560	1666	192	9						82064
10	1023	14362	54972	93104	83790	43596	13524	2464	243	10					307088
11	2047	32796	147087	298464	329350	216216	88102	22464	3483	300	11				1140320
12	4095	73758	380511	905300	1193550	956676	489510	163152	35235	4750	363	12			4206912
13	8191	163872	957996	2626320	4053575	3869856	2408448	1003968	282285	52800	6292	432	13		15434048
14	16383	360482	2358774	7345688	13056525	14560506	10761156	5438112	1905849	462550	76230	8136	507	14	56350912

Technical details

Node id: n-6f13d958f71d4ca4 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-9b3d5c5394a7d2f3

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:   3,    2;
  3:   7,   12,     3;
  4:  15,   46,    27,     4;
  5:  31,  144,   144,    48,     5;
  6:  63,  402,   594,   328,    75,    6;
  7: 127, 1044,  2097,  1680,   625,  108,    7;
  8: 255, 2582,  6669,  7180,  3825, 1062,  147,   8;
  9: 511, 6168, 19686, 27072, 19200, 7560, 1666, 192, 9;

%F Formula

The row polynomials satisfy P(n,x) = -4*P(n-4,x) + (12+4*x)*P(n-3,x) - (13+6*x+x^2)*P(n-2,x) + (6+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 3 + 2*x, P(3,x) = 7 + 12*x + 3*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[3, x_] := 7 + 12*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-4*P[n - 4, x] + (12 + 4*x)*P[n - 3, x] - (13 + 6*x + x^2)*P[n - 2, x] + (6 + 2*x)*P[n - 1, 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; v[3] = 3 + 2*x; v[4] = 7 + 12*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = -4*v[i - 4] + (12 + 4*x)*v[i - 3] - (13 + 6*x + x^2)*v[i - 2] + (6 + 2*x)*v[i - 1]); 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 Riordan array (1, x/(1-3*x+2*x^2)).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Related ?

Same fingerprint ?

A110441 → θ+1

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.