A113214 → x ↦ x−2

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	1														1
2	-2	-1	1													-2
3	4	-2	-2	1												1
4	-4	8	-1	-3	1											1
5	0	-12	12	1	-4	1										-2
6	8	4	-26	15	4	-5	1									1
7	-16	24	20	-46	16	8	-6	1								1
8	16	-64	40	56	-71	14	13	-7	1							-2
9	0	80	-160	40	120	-99	8	19	-8	1						1
10	-32	-16	256	-312	-2	219	-127	-3	26	-9	1					1
11	64	-160	-128	640	-508	-122	358	-151	-20	34	-10	1				-2
12	-64	384	-432	-528	1348	-704	-365	539	-166	-44	43	-11	1			1
13	0	-448	1344	-784	-1568	2492	-812	-783	760	-166	-76	53	-12	1		1
14	128	64	-1888	3536	-872	-3796	4142	-689	-1432	1014	-144	-117	64	-13	1	-2

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Technical details

Node id: n-317422c875f06510 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint: f6-2a6d8930c1d2f707

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:   0,   1;
  2:  -2,  -1,    1;
  3:   4,  -2,   -2,   1;
  4:  -4,   8,   -1,  -3,   1;
  5:   0, -12,   12,   1,  -4,   1;
  6:   8,   4,  -26,  15,   4,  -5,  1;
  7: -16,  24,   20, -46,  16,   8, -6,  1;
  8:  16, -64,   40,  56, -71,  14, 13, -7,  1;
  9:   0,  80, -160,  40, 120, -99,  8, 19, -8, 1;

%F Formula

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

%t Mathematica

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

by the cited steps: e-e475554f8902db13
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-e475554f8902db13

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-e475554f8902db13

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1+2x,x(1+x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
  - theorem_check ma_wang_interval e-5a8f629e61d609b9 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-e5bdc121bd99907c basis_map (A038207) → Riordan array (1+2x,x(1+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