A099093 → x ↦ x−1

Proved for all n: all roots real, in (−1/3, 1] · interlacing (weak) what this means

Recurrence

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

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

\(P_{1}(x) = -3 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-3	3														0
2	6	-15	9													0
3	-9	45	-63	27												0
4	9	-99	252	-243	81											0
5	0	162	-729	1215	-891	243										0
6	-27	-162	1620	-4347	5346	-3159	729									0
7	81	-81	-2673	12069	-22761	22113	-10935	2187								0
8	-162	891	2430	-26325	75411	-109107	87480	-37179	6561							0
9	243	-2673	3159	42039	-200718	418932	-490617	334611	-124659	19683						0
10	-243	5589	-22113	-30375	423063	-1305396	2138886	-2101707	1246590	-413343	59049					0
11	0	-8748	65610	-91854	-632043	3326427	-7604199	10246095	-8667081	4546773	-1358127	177147				0
12	729	8748	-139968	497178	260253	-6690033	22459032	-40829103	46694637	-34661763	16297524	-4428675	531441			0
13	-2187	2187	223074	-1439046	2331342	8975448	-54655317	136313523	-205831692	204427638	-135163161	57572775	-14348907	1594323		0
14	4374	-37179	-216513	3074922	-10600389	918540	103445100	-383042115	763864425	-986708790	865894536	-516029211	200884698	-46235367	4782969	0

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Technical details

Node id: n-c7f2f3338041a8d8 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099093
Fingerprint: f6-745478c8d38f3351

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): no
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    1;
  1:   -3,     3;
  2:    6,   -15,     9;
  3:   -9,    45,   -63,     27;
  4:    9,   -99,   252,   -243,      81;
  5:    0,   162,  -729,   1215,    -891,     243;
  6:  -27,  -162,  1620,  -4347,    5346,   -3159,     729;
  7:   81,   -81, -2673,  12069,  -22761,   22113,  -10935,   2187;
  8: -162,   891,  2430, -26325,   75411, -109107,   87480, -37179,    6561;
  9:  243, -2673,  3159,  42039, -200718,  418932, -490617, 334611, -124659, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(3-3*x)*P(n-1,x) - (3-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -3 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -3 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(3 - 3*x)*P[n - 1, x] - (3 - 3*x)*P[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] = 1; v[2] = -3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(3 - 3*x)*v[i - 1] - (3 - 3*x)*v[i - 2]); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-1/3, 1]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-9ac4433d08f91ae9
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-9ac4433d08f91ae9

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image affine, shift 0 e-9ac4433d08f91ae9

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1, 3+3x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
  - theorem_check ma_wang_interval e-a3711e6f8037cbc8 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-344ac1c74a22d15f basis_map (A007318) → Riordan array (1, 3+3x).

Related sequences ?

No candidate equalities or related sequences.

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741