A099093 → x ↦ x−4

Proved for all n: all roots real, in (8/3, 4] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-12 + 3\,x\right)\,P_{n-1}(x) + \left(-12 + 3\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -12 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-12	3														-9
2	132	-69	9													72
3	-1440	1152	-306	27												-567
4	15696	-16920	6813	-1215	81											4455
5	-171072	231984	-125388	33777	-4536	243										-34992
6	1864512	-3046896	2068092	-746469	151146	-16281	729									274833
7	-20321280	38859264	-31757184	14380416	-3897396	632286	-56862	2187								-2158569
8	221481216	-485118720	463706208	-252674640	85856841	-18630810	2521611	-194643	6561							16953624
9	-2413919232	5958593280	-6522166656	4155377760	-1698396012	461860623	-83572560	9703719	-656100	19683						-133155495
10	26309256192	-72259008768	89121949056	-65007818784	31058579412	-10156375269	2302300827	-357261759	36321696	-2184813	59049					1045816839
11	-286744043520	867290996736	-1190098635264	978028639488	-534879523872	204414725952	-55708283142	10826881281	-1470661272	132978348	-7203978	177147				-8213952096
12	3125217447936	-10321688217600	15596816198400	-14259179907072	8784913795632	-3842563041576	1223646839829	-285853381209	48620933478	-5872541148	478119753	-23560551	531441			64513217313
13	-34061680852992	121968186863616	-201243802420224	202593967900416	-138843865063872	68407882602768	-24929707627116	6804133669197	-1390882766256	210325570218	-22869677700	1693348173	-76527504	1594323		-506693386953
14	371237560860672	-1432567373967360	2562703330599936	-2816958414585600	2125711779198912	-1164960688539024	478168688131020	-149337545817717	35661982857300	-6480221847174	882057782574	-87212125305	5921315622	-247120065	4782969	3979621526760

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Technical details

Node id: n-e58e3554d9394f92 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099093
Fingerprint: f6-15c1d58202ae4165

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:         -12,          3;
  2:         132,        -69,           9;
  3:       -1440,       1152,        -306,         27;
  4:       15696,     -16920,        6813,      -1215,          81;
  5:     -171072,     231984,     -125388,      33777,       -4536,       243;
  6:     1864512,   -3046896,     2068092,    -746469,      151146,    -16281,       729;
  7:   -20321280,   38859264,   -31757184,   14380416,    -3897396,    632286,    -56862,    2187;
  8:   221481216, -485118720,   463706208, -252674640,    85856841, -18630810,   2521611, -194643,    6561;
  9: -2413919232, 5958593280, -6522166656, 4155377760, -1698396012, 461860623, -83572560, 9703719, -656100, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(12-3*x)*P(n-1,x) - (12-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -12 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -12 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(12 - 3*x)*P[n - 1, x] - (12 - 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] = -12 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(12 - 3*x)*v[i - 1] - (12 - 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 (8/3, 4]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-4e1348328210bdec
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-4e1348328210bdec
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-4e1348328210bdec

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-4e1348328210bdec

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-3547903a43c489fd basis_map (A038231) → 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