A099097 → x ↦ x−9

Proved for all n: all roots real, in (77/9, 9] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-27 + 3\,x\right)\,P_{n-1}(x) + \left(-9 + x\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = -27 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-27	3							-24
3	720	-161	9						568
4	-19197	6453	-723	27					-13440
5	511839	-229653	38638	-2889	81				318016
6	-13646880	7658874	-1719225	192951	-10827	243			-7524864
7	363859209	-245151522	68818302	-10302713	867564	-38961	729		178052608
8	-9701376723	7628091975	-2570416821	481172373	-54041973	3641625	-136323	2187	-4213063680

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ebdb57c5e69f5463 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint: f6-b1d0161dfb5220fa

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:            -27,             3;
   3:            720,          -161,              9;
   4:         -19197,          6453,           -723,           27;
   5:         511839,       -229653,          38638,        -2889,            81;
   6:      -13646880,       7658874,       -1719225,       192951,        -10827,         243;
   7:      363859209,    -245151522,       68818302,    -10302713,        867564,      -38961,         729;
   8:    -9701376723,    7628091975,    -2570416821,    481172373,     -54041973,     3641625,     -136323,     2187;
   9:   258662438640, -232492390587,    91421013852, -20541361815,    2884539601,  -259231581,    14560074,  -467289,     6561;
  10: -6896573452773, 6974927657271, -3135082702401, 821978842383, -138539104542, 15566054892, -1165948209, 56141019, -1576827, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(27-3*x)*P(n-1,x) - (9-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -27 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -27 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(27 - 3*x)*P[n - 1, x] - (9 - 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] = -27 + 3*x; for(n = 3, N, my(i = n); v[i] = -(27 - 3*x)*v[i - 1] - (9 - 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 (77/9, 9]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-0c114df991087062
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-0c114df991087062
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-0c114df991087062

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-0c114df991087062

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-2d0268d4f6f3cbee basis_map (A038291) → 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