A113214 → x ↦ x−7

Proved for all n: all roots real, in (3, 7] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-7 + x\right)\,P_{n-1}(x) + \left(-7 + x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -5 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-5	1														-4
2	28	-11	1													18
3	-161	93	-17	1												-84
4	931	-707	194	-23	1											396
5	-5390	5068	-1853	331	-29	1										-1872
6	31213	-34986	15974	-3815	504	-35	1									8856
7	-180761	235249	-128765	38509	-6809	713	-41	1								-41904
8	1046836	-1551389	989800	-355649	78829	-11051	958	-47	1							198288
9	-6062525	10079055	-7343385	3081015	-821280	144386	-16757	1239	-53	1						-938304
10	35109823	-64709351	53002761	-25431147	7922523	-1675796	243928	-24143	1556	-59	1					4440096
11	-203331086	411459370	-374245928	202110300	-72058833	17821113	-3121607	387499	-33425	1909	-65	1				-21010752
12	1177548841	-2595471396	2595452188	-1557997238	625633323	-177153529	36289070	-5421171	586439	-44819	2298	-71	1			99423936
13	-6819524285	16262301937	-17732455846	11712414826	-5230908368	1668901402	-391504657	68403167	-8904770	853384	-58541	2723	-77	1		-470479104
14	39493828108	-101309789231	119625856147	-86217926774	42391342903	-15047510156	3978256982	-796089559	121210313	-13978286	1202266	-74807	3184	-83	1	2226331008

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Technical details

Node id: n-2ae85de8aa2f7a20 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint: f6-03f32002a7d4b1e9

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:       -5,        1;
  2:       28,      -11,        1;
  3:     -161,       93,      -17,       1;
  4:      931,     -707,      194,     -23,       1;
  5:    -5390,     5068,    -1853,     331,     -29,      1;
  6:    31213,   -34986,    15974,   -3815,     504,    -35,      1;
  7:  -180761,   235249,  -128765,   38509,   -6809,    713,    -41,    1;
  8:  1046836, -1551389,   989800, -355649,   78829, -11051,    958,  -47,   1;
  9: -6062525, 10079055, -7343385, 3081015, -821280, 144386, -16757, 1239, -53, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(7-x)*P(n-1,x) - (7-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -5 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -5 + x;
P[n_, x_] := P[n, x] = Expand[-(7 - x)*P[n - 1, x] - (7 - 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] = -5 + x; for(n = 2, N, my(i = n + 1); v[i] = -(7 - x)*v[i - 1] - (7 - 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 (3, 7]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-921246387fcf3a1f
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-921246387fcf3a1f
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-921246387fcf3a1f

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-921246387fcf3a1f

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-9b73612a2d328481 basis_map (A027466) → 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