A113953 → x ↦ x−1

Proved for all n: all roots real, in (−7, 1] · interlacing (weak) what this means

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-1	1														0
2	-1	0	1													0
3	3	-5	1	1												0
4	-1	6	-8	2	1											0
5	-5	9	2	-10	3	1										0
6	7	-28	35	-8	-11	4	1									0
7	3	7	-49	67	-23	-11	5	1								0
8	-17	66	-70	-30	96	-42	-10	6	1							0
9	11	-91	248	-272	54	114	-64	-8	7	1						0
10	23	-64	-67	440	-578	216	114	-88	-5	8	1					0
11	-45	291	-675	533	366	-914	458	90	-113	-1	9	1				0
12	-1	-162	972	-2222	2203	-308	-1168	772	37	-138	4	10	1			0
13	91	-511	798	778	-4091	5071	-1884	-1204	1141	-49	-162	10	11	1		0
14	-89	924	-3577	6408	-3981	-4140	8675	-4560	-875	1540	-171	-184	17	12	1	0

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Technical details

Node id: n-86ae49c0879d58f0 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113953
Fingerprint: f6-045939b799da74a9

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:  -1,   1;
  2:  -1,   0,   1;
  3:   3,  -5,   1,    1;
  4:  -1,   6,  -8,    2,   1;
  5:  -5,   9,   2,  -10,   3,   1;
  6:   7, -28,  35,   -8, -11,   4,   1;
  7:   3,   7, -49,   67, -23, -11,   5,  1;
  8: -17,  66, -70,  -30,  96, -42, -10,  6, 1;
  9:  11, -91, 248, -272,  54, 114, -64, -8, 7, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[-(1 - x)*P[n - 1, x] - (2 - 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] = -1 + x; for(n = 2, N, my(i = n + 1); v[i] = -(1 - x)*v[i - 1] - (2 - 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 (-7, 1]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-38de1ebb8f2e73b6
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-38de1ebb8f2e73b6

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-38de1ebb8f2e73b6

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-89ed187733702f47 basis_map (A007318) → 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