A113953 → x ↦ x−2

Proved for all n: all roots real, in (−6, 2] · interlacing (weak) what this means

Recurrence

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

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

\(P_{1}(x) = -2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-2	1							-1
2	0	-2	1						-1
3	8	-4	-2	1					3
4	-16	24	-8	-2	1				-1
5	0	-32	40	-12	-2	1			-5
6	64	-64	-32	56	-16	-2	1		7
7	-128	320	-224	-16	72	-20	-2	1	3

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-7914dcf2bf97e042 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113953
Fingerprint: f6-3261a32f89c887ba

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 ?

A001607: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS

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:   -2,    1;
  2:    0,   -2,    1;
  3:    8,   -4,   -2,    1;
  4:  -16,   24,   -8,   -2,    1;
  5:    0,  -32,   40,  -12,   -2,   1;
  6:   64,  -64,  -32,   56,  -16,  -2,   1;
  7: -128,  320, -224,  -16,   72, -20,  -2,   1;
  8:    0, -384,  768, -480,   16,  88, -24,  -2,  1;
  9:  512, -768, -384, 1344, -832,  64, 104, -28, -2, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -2 + x;
P[n_, x_] := P[n, x] = Expand[-(2 - x)*P[n - 1, x] - (4 - 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] = -2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(2 - x)*v[i - 1] - (4 - 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 (-6, 2]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-c345fe1d5254381d
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-c345fe1d5254381d

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-c345fe1d5254381d

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-a0ae771232cbba16 basis_map (A038207) → 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