A099093 → x ↦ x−6

Proved for all n: all roots real, in (14/3, 6] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-18 + 3\,x\right)\,P_{n-1}(x) + \left(-18 + 3\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -18 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-18	3														-15
2	306	-105	9													210
3	-5184	2700	-468	27												-2925
4	87804	-61344	16047	-1863	81											40725
5	-1487160	1303452	-456354	79785	-6966	243										-567000
6	25188408	-26556012	11651850	-2723517	357696	-25029	729									7894125
7	-426622464	525649824	-277276608	81173664	-14244336	1498338	-87480	2187								-109906875
8	7225813008	-10187990784	6278527080	-2208976920	485309961	-68179482	5981445	-299619	6561							1530191250
9	-122385429792	194319708912	-137009531376	56304210024	-14862591018	2613457467	-306134802	23035671	-1010394	19683						-21304265625
10	2072873102112	-3659849776656	2905553231712	-1365907208760	421076758338	-88946846901	13038594381	-1309709007	86277150	-3365793	59049					296611115625
11	-35108778101760	68231004236352	-60230376809280	31878485078256	-11240664007968	2772643511772	-488184440724	61357498785	-5394821616	316030248	-11101212	177147				-4129602750000
12	594646289993664	-1261388495273472	1225560287775312	-721200872383632	286290304101828	-80765301716568	16603715228787	-2506297755033	275697169722	-21553593588	1136752299	-36315135	531441			57494874515625
13	-10071675214054272	23155447374343872	-24555410870499936	15903792704394864	-7018860683605608	2229016767967908	-524057528799522	92355516976653	-12200263034652	1193183184438	-83974409586	4027437045	-117979902	1594323		-800479076484375
14	170586520633090944	-422524146595449024	485619487126254432	-343276204724376048	166734042327101736	-58866237531035460	15578923043027250	-3139647386701365	484189843233600	-56863030230090	5005966603716	-320353166241	14087438028	-381043197	4782969	11144763029531250

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Technical details

Node id: n-4c459b8c0c09c21c · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099093
Fingerprint: f6-c4b95ff4d415b654

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:           -18,            3;
  2:           306,         -105,             9;
  3:         -5184,         2700,          -468,          27;
  4:         87804,       -61344,         16047,       -1863,           81;
  5:      -1487160,      1303452,       -456354,       79785,        -6966,        243;
  6:      25188408,    -26556012,      11651850,    -2723517,       357696,     -25029,        729;
  7:    -426622464,    525649824,    -277276608,    81173664,    -14244336,    1498338,     -87480,     2187;
  8:    7225813008, -10187990784,    6278527080, -2208976920,    485309961,  -68179482,    5981445,  -299619,     6561;
  9: -122385429792, 194319708912, -137009531376, 56304210024, -14862591018, 2613457467, -306134802, 23035671, -1010394, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(18-3*x)*P(n-1,x) - (18-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -18 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -18 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(18 - 3*x)*P[n - 1, x] - (18 - 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] = -18 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(18 - 3*x)*v[i - 1] - (18 - 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 (14/3, 6]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-6c0da882dbe0cffe
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-6c0da882dbe0cffe
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-6c0da882dbe0cffe

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-6c0da882dbe0cffe

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-e9cb24f9617f2dae basis_map (A038255) → 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