A099040 → x ↦ x−8

Proved for all n: all roots real, in (6, 8] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-16 + 2\,x\right)\,P_{n-1}(x) + \left(-16 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -16 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-16	2														-14
2	240	-62	4													182
3	-3584	1408	-184	8												-2352
4	53504	-28224	5572	-488	16											30380
5	-798720	528896	-139840	18456	-1216	32										-392392
6	11923456	-9501184	3149632	-556024	55136	-2912	64									5068168
7	-177995776	165806080	-66101248	14620672	-1937856	153920	-6784	128								-65460864
8	2657157120	-2833022976	1319835648	-350937600	58252816	-6181568	409536	-15488	256							845497744
9	-39666581504	47633793024	-25394184192	7888539648	-1573673216	209072288	-18499328	1051264	-34816	512						-10920516320
10	592150790144	-790831169536	474791116800	-168750329856	39321930496	-6277092320	695218112	-52752000	2624512	-77312	1024					141050260064
11	-8839747338240	12996126441472	-8676745674752	3472582508544	-925695696896	172584835072	-22963540608	2180649344	-144836608	6408192	-169984	2048				-1821816412416
12	131961544769536	-211779917447168	155641863471104	-69265223974912	20789644619776	-4433671416832	688908645440	-78583002496	6531188224	-385718272	15365120	-370688	4096			23530726132928
13	-1969948758900736	3426784250953728	-2749009466753024	1346612499054592	-449408465698816	107905283153920	-19177294840832	2554327860096	-254986332160	18841664512	-1001742336	36288512	-802816	8192		-303924736087168
14	29407795426099200	-55116043764367360	47923890319523840	-25624291607838720	9412595687424000	-2512783429951488	502757402600448	-76588690112384	8926772018176	-792205427712	52693927936	-2547439616	84615168	-1728512	16384	3925516139359360

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Technical details

Node id: n-e10282c2431b31a3 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-0ab0161c248d2b80

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:          -16,           2;
  2:          240,         -62,            4;
  3:        -3584,        1408,         -184,          8;
  4:        53504,      -28224,         5572,       -488,          16;
  5:      -798720,      528896,      -139840,      18456,       -1216,        32;
  6:     11923456,    -9501184,      3149632,    -556024,       55136,     -2912,        64;
  7:   -177995776,   165806080,    -66101248,   14620672,    -1937856,    153920,     -6784,     128;
  8:   2657157120, -2833022976,   1319835648, -350937600,    58252816,  -6181568,    409536,  -15488,    256;
  9: -39666581504, 47633793024, -25394184192, 7888539648, -1573673216, 209072288, -18499328, 1051264, -34816, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(16-2*x)*P(n-1,x) - (16-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -16 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -16 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(16 - 2*x)*P[n - 1, x] - (16 - 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] = -16 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16 - 2*x)*v[i - 1] - (16 - 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, 8]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-4478346993b2c814
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-4478346993b2c814
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-4478346993b2c814

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-4478346993b2c814

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-8232f658d776d13c basis_map (A038279) → 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