A099093 → x ↦ x−9

Proved for all n: all roots real, in (23/3, 9] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-27 + 3\,x\right)\,P_{n-1}(x) + \left(-27 + 3\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -27 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-27	3							-24
2	702	-159	9						552
3	-18225	6237	-711	27					-12672
4	473121	-216675	37188	-2835	81				290880
5	-12282192	7046514	-1616193	185247	-10611	243			-6676992
6	318844917	-219832866	63122652	-9662139	831546	-38151	729		153266688
7	-8277193575	6664919679	-2299033449	440395461	-50595921	3486321	-133407	2187	-3518152704

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-767aa1cf87a969b2 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099093
Fingerprint: f6-d9ff792219ed8e46

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:            -27,             3;
  2:            702,          -159,              9;
  3:         -18225,          6237,           -711,           27;
  4:         473121,       -216675,          37188,        -2835,            81;
  5:      -12282192,       7046514,       -1616193,       185247,        -10611,         243;
  6:      318844917,    -219832866,       63122652,     -9662139,        831546,      -38151,         729;
  7:    -8277193575,    6664919679,    -2299033449,    440395461,     -50595921,     3486321,     -133407,     2187;
  8:   214875413766, -197892389925,    79704851958, -18337532085,    2635838091,  -242393715,    13926816,  -457083,     6561;
  9: -5578151945157, 5782936357215, -2663639510481, 715440144375, -123492948462, 14206226148, -1089144225, 53662419, -1541835, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(27-3*x)*P(n-1,x) - (27-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -27 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -27 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(27 - 3*x)*P[n - 1, x] - (27 - 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] = -27 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(27 - 3*x)*v[i - 1] - (27 - 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 (23/3, 9]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-7d49a73cf4bf68ad
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-7d49a73cf4bf68ad
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-7d49a73cf4bf68ad

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-7d49a73cf4bf68ad

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-70271c52b20af48c basis_map (A038291) → 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