A099097 → x ↦ x−8

Proved for all n: all roots real, in (68/9, 8] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-24 + 3\,x\right)\,P_{n-1}(x) + \left(-8 + x\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = -24 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-24	3							-21
3	568	-143	9						434
4	-13440	5088	-642	27					-8967
5	318016	-160720	30457	-2565	81				185269
6	-7524864	4757184	-1202904	152073	-9612	243			-3827880
7	178052608	-135143232	42736872	-7207487	683694	-34587	729		79088597
8	-4213063680	3732013056	-1416734208	298770816	-37802148	2869614	-121014	2187	-1634065377

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-0d2d3ff9cbf40a4e · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint: f6-8ff78b9cf59e045e

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:              1;
   2:            -24,             3;
   3:            568,          -143,              9;
   4:         -13440,          5088,           -642,           27;
   5:         318016,       -160720,          30457,        -2565,           81;
   6:       -7524864,       4757184,       -1202904,       152073,        -9612,        243;
   7:      178052608,    -135143232,       42736872,     -7207487,       683694,     -34587,        729;
   8:    -4213063680,    3732013056,    -1416734208,    298770816,    -37802148,    2869614,    -121014,     2187;
   9:    99689107456, -100948305920,    44720621952, -11320305440,   1790886961, -181316790,   11472759,  -414801,     6561;
  10: -2358834069504, 2687757496320, -1361073957888, 402042295680, -76341015384, 9663504783, -815458860, 44234991, -1399680, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(24-3*x)*P(n-1,x) - (8-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -24 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -24 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(24 - 3*x)*P[n - 1, x] - (8 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -24 + 3*x; for(n = 3, N, my(i = n); v[i] = -(24 - 3*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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (68/9, 8]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-7827a087c75bbe15
Coefficients of one sign. Every nonzero row, n >= 1: 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-7827a087c75bbe15
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-7827a087c75bbe15

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-7827a087c75bbe15

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1, 3+x).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
  - theorem_check ma_wang_interval e-09c939a271fe525a root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-f43b3e2d606e8300 basis_map (A038279) → Riordan array (1, 3+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