A099040 → x ↦ x−6

Proved for all n: all roots real, in (4, 6] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-12 + 2\,x\right)\,P_{n-1}(x) + \left(-12 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -12 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-12	2							-10
2	132	-46	4						90
3	-1440	768	-136	8					-800
4	15696	-11280	3028	-360	16				7100
5	-171072	154656	-55728	10008	-896	32			-63000
6	1864512	-2031264	919152	-221176	29856	-2144	64		559000
7	-20321280	25906176	-14114304	4260864	-769856	83264	-4992	128	-4960000

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-7ab73c0941700e0f · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-f8a9eb1806dd3b6f

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:         -12,          2;
  2:         132,        -46,           4;
  3:       -1440,        768,        -136,          8;
  4:       15696,     -11280,        3028,       -360,         16;
  5:     -171072,     154656,      -55728,      10008,       -896,       32;
  6:     1864512,   -2031264,      919152,    -221176,      29856,    -2144,       64;
  7:   -20321280,   25906176,   -14114304,    4260864,    -769856,    83264,    -4992,    128;
  8:   221481216, -323412480,   206091648,  -74866560,   16959376, -2453440,   221376, -11392,    256;
  9: -2413919232, 3972395520, -2898740736, 1231223040, -335485632, 60821152, -7336960, 567936, -25600, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(12-2*x)*P(n-1,x) - (12-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -12 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -12 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(12 - 2*x)*P[n - 1, x] - (12 - 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] = -12 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(12 - 2*x)*v[i - 1] - (12 - 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, 6]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-79e1a4a5e521405c
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-79e1a4a5e521405c
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-79e1a4a5e521405c

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-79e1a4a5e521405c

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-162624a94908f408 basis_map (A038255) → 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