A102662 → reverse → D

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 5 + 2\,x\)

\(P_{3}(x) = 11 + 14\,x + 3\,x^{2}\)

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	5	2													7
3	11	14	3												28
4	17	46	27	4											94
5	23	102	117	44	5										291
6	29	182	339	236	65	6									857
7	35	286	765	844	415	90	7								2442
8	41	414	1467	2316	1765	666	119	8							6796
9	47	566	2517	5292	5715	3282	1001	152	9						18581
10	53	742	3987	10604	15225	12258	5607	1432	189	10					50107
11	59	942	5949	19276	35095	37386	23737	8984	1971	230	11				133640
12	65	1166	8475	32524	72445	97770	81655	42520	13689	2630	275	12			353226
13	71	1414	11637	51756	137195	226818	239337	162968	71631	20030	3421	324	13		926615
14	77	1686	15507	78572	242545	478386	618023	529816	302805	114830	28347	4356	377	14	2415341

Technical details

Node id: n-46eed250c6bead4f · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-729a964c86955068

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:  5,   2;
  3: 11,  14,    3;
  4: 17,  46,   27,    4;
  5: 23, 102,  117,   44,    5;
  6: 29, 182,  339,  236,   65,    6;
  7: 35, 286,  765,  844,  415,   90,    7;
  8: 41, 414, 1467, 2316, 1765,  666,  119,   8;
  9: 47, 566, 2517, 5292, 5715, 3282, 1001, 152, 9;

%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) = 0, P(1,x) = 1, P(2,x) = 5 + 2*x, P(3,x) = 11 + 14*x + 3*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[3, x_] := 11 + 14*x + 3*x^2;
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] = 0; v[2] = 1; v[3] = 5 + 2*x; v[4] = 11 + 14*x + 3*x^2; 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.