A113953 → x ↦ x−4

Proved for all n: all roots real, in (−4, 4] · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-4 + x\right)\,P_{n-1}(x) + \left(-8 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -4 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-4	1														-3
2	8	-6	1													3
3	0	16	-8	1												9
4	-64	0	28	-10	1											-45
5	256	-192	-16	44	-12	1										81
6	-512	896	-352	-56	64	-14	1									27
7	0	-2048	2048	-512	-128	88	-16	1								-567
8	4096	0	-5632	3840	-624	-240	116	-18	1							1539
9	-16384	20480	2048	-12800	6336	-624	-400	148	-20	1						-1215
10	32768	-90112	57344	11264	-25472	9504	-432	-616	184	-22	1					-5589
11	0	196608	-294912	118784	36864	-45824	13184	48	-896	224	-24	1				24057
12	-262144	0	737280	-745472	197632	93184	-76096	17056	928	-1248	268	-26	1			-38637
13	1048576	-1835008	-196608	2179072	-1593344	265216	200448	-118336	20608	2336	-1680	316	-28	1		-28431
14	-2097152	7864320	-6946816	-1474560	5480448	-3004416	258560	385152	-174080	23104	4416	-2200	368	-30	1	317115

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Technical details

Node id: n-45555a84b055028a · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113953
Fingerprint: f6-889b0b97fcbb9aef

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:     -4,     1;
  2:      8,    -6,     1;
  3:      0,    16,    -8,      1;
  4:    -64,     0,    28,    -10,    1;
  5:    256,  -192,   -16,     44,  -12,    1;
  6:   -512,   896,  -352,    -56,   64,  -14,    1;
  7:      0, -2048,  2048,   -512, -128,   88,  -16,   1;
  8:   4096,     0, -5632,   3840, -624, -240,  116, -18,   1;
  9: -16384, 20480,  2048, -12800, 6336, -624, -400, 148, -20, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(4-x)*P(n-1,x) - (8-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -4 + x.

%t Mathematica

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

by the cited steps: e-b26782bfa4993a06
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-b26782bfa4993a06

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-b26782bfa4993a06

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-b4c854eb781c5167 basis_map (A038231) → 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