A099093 → x ↦ x−8

Proved for all n: all roots real, in (20/3, 8] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-24 + 3\,x\right)\,P_{n-1}(x) + \left(-24 + 3\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -24 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-24	3							-21
2	552	-141	9						420
3	-12672	4896	-630	27					-8379
4	290880	-150480	29169	-2511	81				167139
5	-6676992	4328640	-1121688	145233	-9396	243			-3333960
6	153266688	-119434176	38754936	-6702885	651726	-33777	729		66503241
7	-3518152704	3202301952	-1248514560	270283392	-35088876	2731806	-118098	2187	-1326554901

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-b2ac077f0522ccfd · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099093
Fingerprint: f6-d54feed28f6c95b0

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:            -24,             3;
  2:            552,          -141,              9;
  3:         -12672,          4896,           -630,           27;
  4:         290880,       -150480,          29169,        -2511,           81;
  5:       -6676992,       4328640,       -1121688,       145233,        -9396,        243;
  6:      153266688,    -119434176,       38754936,     -6702885,       651726,     -33777,        729;
  7:    -3518152704,    3202301952,    -1248514560,    270283392,    -35088876,    2731806,    -118098,     2187;
  8:    80757264384,  -84083484672,    38282834304,  -9955211040,   1617233121, -168064146,   10910943,  -404595,     6561;
  9: -1853738680320, 2172865720320, -1131467222016, 343541222784, -67026244824, 8714408895, -755025300, 42036327, -1364688, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(24-3*x)*P(n-1,x) - (24-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -24 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -24 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(24 - 3*x)*P[n - 1, x] - (24 - 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] = -24 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(24 - 3*x)*v[i - 1] - (24 - 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 (20/3, 8]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-b4aa148b93b9dcb2
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-b4aa148b93b9dcb2
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-b4aa148b93b9dcb2

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-b4aa148b93b9dcb2

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-78bb889c5d4c7dc1 basis_map (A038279) → 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