A099093 → x ↦ x−15

Proved for all n: all roots real, in (41/3, 15] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-45 + 3\,x\right)\,P_{n-1}(x) + \left(-45 + 3\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -45 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-45	3							-42
2	1980	-267	9						1722
3	-87075	17685	-1197	27					-70560
4	3829275	-1039095	105714	-4779	81				2891196
5	-168399000	57190050	-7767495	527391	-17901	243			-118466712
6	7405637625	-3020502150	513233010	-46502883	2369736	-64395	729		4854171672
7	-325675738125	155060760375	-31635884475	3585293685	-243759051	9942345	-225261	2187	-198899608320

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-8910fc24d8983662 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099093
Fingerprint: f6-6a7f301aedff1f06

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:              -45,               3;
  2:             1980,            -267,                9;
  3:           -87075,           17685,            -1197,             27;
  4:          3829275,        -1039095,           105714,          -4779,             81;
  5:       -168399000,        57190050,         -7767495,         527391,         -17901,          243;
  6:       7405637625,     -3020502150,        513233010,      -46502883,        2369736,       -64395,         729;
  7:    -325675738125,    155060760375,     -31635884475,     3585293685,     -243759051,      9942345,     -225261,      2187;
  8:   14322154522500,  -7796621921625,    1856640090600,  -252613540485,    21478891581,  -1168675695,    39737790,   -772011,     6561;
  9: -629841545296875, 385859688609375, -105049872759375, 16681283724375, -1702665704250, 115848398340, -5254263855, 153179667, -2604717, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(45-3*x)*P(n-1,x) - (45-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -45 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -45 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(45 - 3*x)*P[n - 1, x] - (45 - 3*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] = -45 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(45 - 3*x)*v[i - 1] - (45 - 3*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 (41/3, 15]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-b320272c2500474a
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-b320272c2500474a
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-b320272c2500474a

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-b320272c2500474a

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1, 3+3x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4/3, 0]
  - theorem_check ma_wang_interval e-a3711e6f8037cbc8 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-35fdacf74fcfdf8c basis_map (A027467) → Riordan array (1, 3+3x).

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