A099097 → x ↦ x−3

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

Recurrence

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

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

\(P_{2}(x) = -9 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	-9	3														-6
3	78	-53	9													34
4	-675	693	-237	27												-192
5	5841	-8025	4132	-945	81											1084
6	-50544	86994	-59859	20583	-3537	243										-6120
7	437373	-904662	779292	-357857	92394	-12717	729									34552
8	-3784725	9142551	-9461043	5436981	-1873923	387369	-44469	2187								-195072
9	32750406	-90485775	109334502	-75463095	32541211	-8977545	1547424	-152361	6561							1101328
10	-283399479	881410815	-1217942163	981399375	-508201434	175385508	-40338675	5962491	-513945	19683						-6217824
11	2452344093	-8478688041	13187222631	-12150697077	7344924303	-3043600028	875584782	-172673937	22340934	-1712421	59049					35104288
12	-21220898400	80737592724	-139585830498	144755801298	-100050406281	48392815203	-16714661589	4122593631	-711585891	81861597	-5649021	177147				-198190080
13	183631053321	-762412621500	1450444896720	-1671920389314	1300535590437	-719210831283	289940045536	-85853720853	18532357173	-2844033849	294536412	-18482337	531441			1118931904
14	-1589016784689	7149173076891	-14841746850762	18824764959594	-16275674461734	10129275400968	-4668556103703	1613421181803	-418095025812	80236205478	-11084120595	1043769807	-60052833	1594323		-6317211264
15	13750257902238	-66638739128265	149909493575871	-207481919123970	197381837873763	-136532333909628	70815800168340	-27978957739241	8461667985345	-1949346468018	336738058704	-42296306625	3651708258	-193975965	4782969	35665403776

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Technical details

Node id: n-f75029e1f8101147 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint: f6-025a4bab93003c11

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:          1;
   2:         -9,         3;
   3:         78,       -53,           9;
   4:       -675,       693,        -237,        27;
   5:       5841,     -8025,        4132,      -945,         81;
   6:     -50544,     86994,      -59859,     20583,      -3537,       243;
   7:     437373,   -904662,      779292,   -357857,      92394,    -12717,       729;
   8:   -3784725,   9142551,    -9461043,   5436981,   -1873923,    387369,    -44469,    2187;
   9:   32750406, -90485775,   109334502, -75463095,   32541211,  -8977545,   1547424, -152361,    6561;
  10: -283399479, 881410815, -1217942163, 981399375, -508201434, 175385508, -40338675, 5962491, -513945, 19683;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -9 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(9 - 3*x)*P[n - 1, x] - (3 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -9 + 3*x; for(n = 3, N, my(i = n); v[i] = -(9 - 3*x)*v[i - 1] - (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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (23/9, 3]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-f5b61f1417a02797
Coefficients of one sign. Every nonzero row, n >= 1: 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-f5b61f1417a02797
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-f5b61f1417a02797

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-f5b61f1417a02797

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1, 3+x).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
  - theorem_check ma_wang_interval e-09c939a271fe525a root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-3d5016cd0e3a44e4 basis_map (A027465) → Riordan array (1, 3+x).

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