A099093 → x ↦ x−3

Proved for all n: all roots real, in (5/3, 3] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-9 + 3\,x\right)\,P_{n-1}(x) + \left(-9 + 3\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -9 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-9	3														-6
2	72	-51	9													30
3	-567	621	-225	27												-144
4	4455	-6615	3654	-891	81											684
5	-34992	65610	-48843	18063	-3321	243										-3240
6	274833	-622566	583686	-290115	80676	-11907	729									15336
7	-2158569	5732127	-6484455	4052997	-1512351	337041	-41553	2187								-72576
8	16953624	-51637257	68435604	-51568245	24173721	-7221231	1342818	-142155	6561							343440
9	-133155495	457531335	-695275731	613490679	-346498074	129941820	-32363955	5163507	-478953	19683						-1625184
10	1045816839	-4001652315	6859243377	-6937822287	4586686479	-2071458360	647352000	-138255579	19315584	-1594323	59049					7690464
11	-8213952096	34635072852	-66108071754	75410887410	-57134690469	30194214075	-11359442025	3042792783	-568805895	70681653	-5255361	177147				-36391680
12	64513217313	-297205190604	625139717004	-794004071238	678351231279	-410748813405	180777077370	-58277104911	13659024411	-2270256903	254028798	-17183259	531441			172207296
13	-506693386953	2532028855419	-5818995160506	8144433590202	-7746728418774	5288641016400	-2666422516095	1005361715187	-283514903028	59066832798	-8837686683	899375319	-55801305	1594323		-814893696
14	3979621526760	-21439953492255	53449169985963	-81735432001182	85666683244347	-65106161389440	37004485557510	-15980697808659	5269956738381	-1320736818906	247642648650	-33690702195	3144004956	-180158499	4782969	3856118400

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Technical details

Node id: n-4ec583084451043e · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099093
Fingerprint: f6-4fed01f361177ba4

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:         -9,         3;
  2:         72,       -51,          9;
  3:       -567,       621,       -225,        27;
  4:       4455,     -6615,       3654,      -891,         81;
  5:     -34992,     65610,     -48843,     18063,      -3321,       243;
  6:     274833,   -622566,     583686,   -290115,      80676,    -11907,       729;
  7:   -2158569,   5732127,   -6484455,   4052997,   -1512351,    337041,    -41553,    2187;
  8:   16953624, -51637257,   68435604, -51568245,   24173721,  -7221231,   1342818, -142155,    6561;
  9: -133155495, 457531335, -695275731, 613490679, -346498074, 129941820, -32363955, 5163507, -478953, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(9-3*x)*P(n-1,x) - (9-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -9 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -9 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(9 - 3*x)*P[n - 1, x] - (9 - 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] = -9 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 - 3*x)*v[i - 1] - (9 - 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 (5/3, 3]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-7438d71d24adbb8f
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-7438d71d24adbb8f
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-7438d71d24adbb8f

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-7438d71d24adbb8f

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-69bbc7837aabb73c basis_map (A027465) → 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