A113214 → x ↦ x−15

Proved for all n: all roots real, in (11, 15] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-15 + x\right)\,P_{n-1}(x) + \left(-15 + x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -13 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-13	1														-12
2	180	-27	1													154
3	-2505	557	-41	1												-1988
4	34875	-10275	1130	-55	1											25676
5	-485550	178140	-26053	1899	-69	1										-331632
6	6760125	-2968650	541710	-52583	2864	-83	1									4283384
7	-94118625	48132225	-10525365	1275917	-92609	4025	-97	1								-55324528
8	1310377500	-764812125	194918400	-28333665	2569509	-148875	5382	-111	1							714576016
9	-18243883125	11966457375	-3482575425	590259255	-64211248	4649650	-224125	6935	-125	1						-9229520832
10	254002584375	-184958184375	60516500625	-11716540875	1486551675	-129153364	7781920	-321103	8684	-139	1					119209227424
11	-3536380518750	2830634606250	-1028500605000	223928149500	-32461388025	3289896137	-237870639	12270315	-442553	10629	-153	1				-1539715892288
12	49235669015625	-42967524262500	17165437987500	-4151158233750	676834153875	-78385977945	6612073558	-409326899	18457247	-591219	12770	-167	1			19887093308096
13	-685489327453125	647752633340625	-282190950393750	75045388646250	-13592821572000	1770813992970	-170709125593	12330051679	-667276994	26723544	-769845	15107	-181	1		-256863283821312
14	9543804876562500	-9708030294609375	4580167795171875	-1328438968593750	264634041684375	-38302407643500	4153883795550	-342907923735	21653020985	-1040804622	37488450	-981175	17640	-195	1	3317666667185024

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Technical details

Node id: n-8be1ba22f08cce79 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint: f6-abf295c692fd442c

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:          -13,           1;
  2:          180,         -27,           1;
  3:        -2505,         557,         -41,         1;
  4:        34875,      -10275,        1130,       -55,         1;
  5:      -485550,      178140,      -26053,      1899,       -69,       1;
  6:      6760125,    -2968650,      541710,    -52583,      2864,     -83,       1;
  7:    -94118625,    48132225,   -10525365,   1275917,    -92609,    4025,     -97,    1;
  8:   1310377500,  -764812125,   194918400, -28333665,   2569509, -148875,    5382, -111,    1;
  9: -18243883125, 11966457375, -3482575425, 590259255, -64211248, 4649650, -224125, 6935, -125, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(15-x)*P(n-1,x) - (15-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -13 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -13 + x;
P[n_, x_] := P[n, x] = Expand[-(15 - x)*P[n - 1, x] - (15 - 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] = -13 + x; for(n = 2, N, my(i = n + 1); v[i] = -(15 - x)*v[i - 1] - (15 - 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 (11, 15]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-1b92f65567f189e7
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-1b92f65567f189e7
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-1b92f65567f189e7

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-1b92f65567f189e7

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-71209083fd4bb877 basis_map (A027467) → 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