A099097 → x ↦ x−2

Proved for all n: all roots real, in (14/9, 2] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-6 + 3\,x\right)\,P_{n-1}(x) + \left(-2 + x\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = -6 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	-6	3														-3
3	34	-35	9													8
4	-192	300	-156	27												-21
5	1084	-2272	1783	-621	81											55
6	-6120	16092	-16902	8865	-2322	243										-144
7	34552	-109284	143850	-100871	39744	-8343	729									377
8	-195072	721056	-1141056	1002144	-527568	166482	-29160	2187								-987
9	1101328	-4658432	8612520	-9090440	5991481	-2525166	664605	-99873	6561							2584
10	-6217824	29617392	-62647248	77234856	-61162926	32264907	-11338326	2559519	-336798	19683						-6765
11	35104288	-185939632	442852192	-624557480	577608722	-366036407	160970301	-48507741	9586350	-1121931	59049					17711
12	-198190080	1155498048	-3060020160	4858784496	-5139764064	3803351868	-2008989468	757499985	-199808208	35114472	-3700404	177147				-46368
13	1118931904	-7120574976	20754971088	-36640800304	43635162948	-36929721864	22776015403	-10313982531	3403668762	-798281244	126305811	-12105045	531441			121393
14	-6317211264	43571059392	-138616013088	269332125936	-356595065976	339737352228	-239623927206	126687951957	-50206843764	14730656598	-3110163318	447493005	-39326634	1594323		-317811
15	35665403776	-265017908288	913778739552	-1937804223184	2823655647464	-2990714704620	2374473867250	-1435595512895	664183598400	-234004239630	61802056836	-11864932083	1565270892	-127014399	4782969	832040

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Technical details

Node id: n-0a274609716903e8 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint: f6-fbaeed9a78101bba

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:       -6,        3;
   3:       34,      -35,         9;
   4:     -192,      300,      -156,       27;
   5:     1084,    -2272,      1783,     -621,        81;
   6:    -6120,    16092,    -16902,     8865,     -2322,      243;
   7:    34552,  -109284,    143850,  -100871,     39744,    -8343,       729;
   8:  -195072,   721056,  -1141056,  1002144,   -527568,   166482,    -29160,    2187;
   9:  1101328, -4658432,   8612520, -9090440,   5991481, -2525166,    664605,  -99873,    6561;
  10: -6217824, 29617392, -62647248, 77234856, -61162926, 32264907, -11338326, 2559519, -336798, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(6-3*x)*P(n-1,x) - (2-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -6 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -6 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(6 - 3*x)*P[n - 1, x] - (2 - 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] = -6 + 3*x; for(n = 3, N, my(i = n); v[i] = -(6 - 3*x)*v[i - 1] - (2 - 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 (14/9, 2]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-fbe1f1ba82c385a7
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-fbe1f1ba82c385a7
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-fbe1f1ba82c385a7

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-fbe1f1ba82c385a7

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-33066b18a08d7c1d basis_map (A038207) → 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