A099040 → x ↦ x−1

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-2	2														0
2	2	-6	4													0
3	0	8	-16	8												0
4	-4	0	28	-40	16											0
5	8	-24	-8	88	-96	32										0
6	-8	56	-88	-56	256	-224	64									0
7	0	-64	256	-256	-256	704	-512	128								0
8	16	0	-352	960	-624	-960	1856	-1152	256							0
9	-32	160	64	-1600	3168	-1248	-3200	4736	-2560	512						0
10	32	-352	896	704	-6368	9504	-1728	-9856	11776	-5632	1024					0
11	0	384	-2304	3712	4608	-22912	26368	384	-28672	28672	-12288	2048				0
12	-64	0	2880	-11648	12352	23296	-76096	68224	14848	-79872	68608	-26624	4096			0
13	128	-896	-384	17024	-49792	33152	100224	-236672	164864	74752	-215040	161792	-57344	8192		0
14	-128	1920	-6784	-5760	85632	-187776	64640	385152	-696320	369664	282624	-563200	376832	-122880	16384	0

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Technical details

Node id: n-5b8743d4a9a28750 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-d1cb5a6c597b1ee5

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:  -2,   2;
  2:   2,  -6,    4;
  3:   0,   8,  -16,     8;
  4:  -4,   0,   28,   -40,   16;
  5:   8, -24,   -8,    88,  -96,    32;
  6:  -8,  56,  -88,   -56,  256,  -224,    64;
  7:   0, -64,  256,  -256, -256,   704,  -512,   128;
  8:  16,   0, -352,   960, -624,  -960,  1856, -1152,   256;
  9: -32, 160,   64, -1600, 3168, -1248, -3200,  4736, -2560, 512;

%F Formula

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

%t Mathematica

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

by the cited steps: e-dc7a132d665751c2
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-dc7a132d665751c2

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-dc7a132d665751c2

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1, 2+2x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
  - theorem_check ma_wang_interval e-17786677d2ed021b root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-e158703ce212fb71 basis_map (A007318) → Riordan array (1, 2+2x).

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