A099097 → x ↦ x−4

Proved for all n: all roots real, in (32/9, 4] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-12 + 3\,x\right)\,P_{n-1}(x) + \left(-4 + x\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = -12 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-12	3							-9
3	140	-71	9						78
4	-1632	1248	-318	27					-675
5	19024	-19448	7453	-1269	81				5841
6	-221760	283824	-145260	37161	-4752	243			-50544
7	2585024	-3974352	2545332	-869183	166914	-17091	729		437373
8	-30133248	54090240	-41602176	17772288	-4554348	700110	-59778	2187	-3784725

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-943ef8677f31ba50 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint: f6-fc3e96be02f30e16

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:         -12,          3;
   3:         140,        -71,           9;
   4:       -1632,       1248,        -318,         27;
   5:       19024,     -19448,        7453,      -1269,          81;
   6:     -221760,     283824,     -145260,      37161,       -4752,       243;
   7:     2585024,   -3974352,     2545332,    -869183,      166914,    -17091,       729;
   8:   -30133248,   54090240,   -41602176,   17772288,    -4554348,    700110,    -59778,     2187;
   9:   351258880, -721000192,   647341152, -332051920,   106432201, -21829086,   2797659,  -204849,    6561;
  10: -4094573568, 9459284736, -9710595456, 5813955168, -2237352492, 573890847, -98119944, 10782639, -691092, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(12-3*x)*P(n-1,x) - (4-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -12 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -12 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(12 - 3*x)*P[n - 1, x] - (4 - 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] = -12 + 3*x; for(n = 3, N, my(i = n); v[i] = -(12 - 3*x)*v[i - 1] - (4 - 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 (32/9, 4]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-27282dcb1ed61456
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-27282dcb1ed61456
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-27282dcb1ed61456

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-27282dcb1ed61456

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-f299bbf6486ea9ca basis_map (A038231) → 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