A113214 → x ↦ x−8

Proved for all n: all roots real, in (4, 8] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-8 + x\right)\,P_{n-1}(x) + \left(-8 + x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -6 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-6	1														-5
2	40	-13	1													28
3	-272	130	-20	1												-161
4	1856	-1168	269	-27	1											931
5	-12672	9888	-3030	457	-34	1										-5390
6	86528	-80576	30808	-6201	694	-41	1									31213
7	-590848	639360	-292912	73730	-11024	980	-48	1								-180761
8	4034560	-4974592	2655616	-802336	150169	-17842	1315	-55	1							1046836
9	-27549696	38125568	-23236864	8191552	-1841766	274041	-26998	1699	-62	1						-6062525
10	188121088	-288722944	197800960	-79694976	20921992	-3741189	461663	-38835	2132	-69	1					35109823
11	-1284571136	2165350400	-1647110144	746591488	-224145232	46817410	-6944468	731753	-53696	2614	-76	1				-203331086
12	8771600384	-16109469696	13471100928	-6784481280	2292682432	-547833008	94938661	-12026149	1105430	-71924	3145	-83	1			1177548841
13	-59896233984	119039983616	-108536045568	60127109120	-22586187392	6076661984	-1204969142	178349361	-19708268	1606214	-93862	3725	-90	1		-6819524285
14	408997068800	-874568744960	863450071040	-521805967360	215690667520	-64524136768	14409072824	-2440616177	315145916	-30877158	2260026	-119853	4354	-97	1	39493828108

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Technical details

Node id: n-e58c4cf54cd7c009 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint: f6-896ccc203557ab88

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:        -6,        1;
  2:        40,      -13,         1;
  3:      -272,      130,       -20,       1;
  4:      1856,    -1168,       269,     -27,        1;
  5:    -12672,     9888,     -3030,     457,      -34,      1;
  6:     86528,   -80576,     30808,   -6201,      694,    -41,      1;
  7:   -590848,   639360,   -292912,   73730,   -11024,    980,    -48,    1;
  8:   4034560, -4974592,   2655616, -802336,   150169, -17842,   1315,  -55,   1;
  9: -27549696, 38125568, -23236864, 8191552, -1841766, 274041, -26998, 1699, -62, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(8-x)*P(n-1,x) - (8-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -6 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -6 + x;
P[n_, x_] := P[n, x] = Expand[-(8 - x)*P[n - 1, x] - (8 - 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] = -6 + x; for(n = 2, N, my(i = n + 1); v[i] = -(8 - x)*v[i - 1] - (8 - 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, 8]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-fbfad365d270ad64
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-fbfad365d270ad64
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-fbfad365d270ad64

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-fbfad365d270ad64

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-4738d2d3e61fb87a basis_map (A038279) → 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