A102662 → reverse → D+1

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1\)

\(P_{1}(x) = 4 + x\)

\(P_{2}(x) = 8 + 7\,x + x^{2}\)

\(P_{3}(x) = 14 + 25\,x + 10\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	1														5
2	8	7	1													16
3	14	25	10	1												50
4	20	63	50	13	1											147
5	26	125	168	83	16	1										419
6	32	211	430	349	124	19	1									1166
7	38	321	908	1099	626	173	22	1								3188
8	44	455	1674	2805	2344	1019	230	25	1							8597
9	50	613	2800	6131	7038	4425	1548	295	28	1						22929
10	56	795	4358	11933	17876	15303	7650	2233	368	31	1					60604
11	62	1001	6420	21259	39914	44405	29968	12375	3094	449	34	1				158982
12	68	1231	9058	35349	80576	112259	97950	54185	19004	4151	538	37	1			414407
13	74	1485	12344	55635	150134	254257	277140	197159	92002	27989	5424	635	40	1		1074319
14	80	1763	16350	83741	262188	526895	697754	618105	369032	148475	39830	6933	740	43	1	2771930

Technical details

Node id: n-d885ef35a1c653b3 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-198071281c691a50

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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:  1;
  1:  4,   1;
  2:  8,   7,    1;
  3: 14,  25,   10,    1;
  4: 20,  63,   50,   13,    1;
  5: 26, 125,  168,   83,   16,    1;
  6: 32, 211,  430,  349,  124,   19,    1;
  7: 38, 321,  908, 1099,  626,  173,   22,   1;
  8: 44, 455, 1674, 2805, 2344, 1019,  230,  25,  1;
  9: 50, 613, 2800, 6131, 7038, 4425, 1548, 295, 28, 1;

%F Formula

The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*x+2*x^2)*P(n-3,x) - (1+x^2)*P(n-2,x) + (2+2*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + x, P(2,x) = 8 + 7*x + x^2, P(3,x) = 14 + 25*x + 10*x^2 + x^3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + x;
P[2, x_] := 8 + 7*x + x^2;
P[3, x_] := 14 + 25*x + 10*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*x + 2*x^2)*P[n - 3, x] - (1 + x^2)*P[n - 2, x] + (2 + 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] = 1; v[2] = 4 + x; v[3] = 8 + 7*x + x^2; v[4] = 14 + 25*x + 10*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - (2*x + 2*x^2)*v[i - 3] - (1 + x^2)*v[i - 2] + (2 + 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from reversal of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).?
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).?
    
    ↑ Liu–Wang theorem with a lag row (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (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.