A113953 → x ↦ x−3

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-3	1														-2
2	3	-4	1													0
3	9	3	-5	1												8
4	-45	30	4	-6	1											-16
5	81	-135	54	6	-7	1										0
6	27	216	-261	80	9	-8	1									64
7	-567	351	405	-429	107	13	-9	1								-128
8	1539	-2862	1134	690	-644	134	18	-10	1							0
9	-1215	6885	-7992	2448	1122	-910	160	24	-11	1						512
10	-5589	-1620	18333	-17208	4326	1760	-1230	184	31	-12	1					-1024
11	24057	-44469	5103	39285	-32022	6750	2670	-1606	205	39	-13	1				0
12	-38637	156006	-173016	27162	74979	-54180	9640	3924	-2039	222	48	-14	1			4096
13	-28431	-191727	555498	-480006	72927	132975	-85620	12844	5599	-2529	234	58	-15	1		-8192
14	317115	-466560	-508113	1486512	-1094337	149040	223635	-128416	16129	7776	-3075	240	69	-16	1	0

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Technical details

Node id: n-0038b0ec9e792748 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113953
Fingerprint: f6-6897785a472f0d9f

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:    -3,     1;
  2:     3,    -4,     1;
  3:     9,     3,    -5,    1;
  4:   -45,    30,     4,   -6,    1;
  5:    81,  -135,    54,    6,   -7,    1;
  6:    27,   216,  -261,   80,    9,   -8,   1;
  7:  -567,   351,   405, -429,  107,   13,  -9,   1;
  8:  1539, -2862,  1134,  690, -644,  134,  18, -10,   1;
  9: -1215,  6885, -7992, 2448, 1122, -910, 160,  24, -11, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(3-x)*P(n-1,x) - (6-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -3 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -3 + x;
P[n_, x_] := P[n, x] = Expand[-(3 - x)*P[n - 1, x] - (6 - 2*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] = -3 + x; for(n = 2, N, my(i = n + 1); v[i] = -(3 - x)*v[i - 1] - (6 - 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 - 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]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-5f9e6216c74dfca3
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-5f9e6216c74dfca3

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-5f9e6216c74dfca3

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): A Jacobsthal triangle.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
  - theorem_check ma_wang_interval e-2d5b16bf2955a407 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-76bf35985f9d5eb4 basis_map (A027465) → A Jacobsthal triangle.

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