A099093 → x ↦ x−12

Proved for all n: all roots real, in (32/3, 12] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-36 + 3\,x\right)\,P_{n-1}(x) + \left(-36 + 3\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -36 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-36	3							-33
2	1260	-213	9						1056
3	-44064	11232	-954	27					-33759
4	1540944	-525096	67077	-3807	81				1079199
5	-53887680	22989744	-3922020	334449	-14256	243			-34499520
6	1884482496	-965767536	206171892	-23467941	1502226	-51273	729		1102870593
7	-65901413376	39431784960	-10109328768	1439555328	-122967396	6300990	-179334	2187	-35256245409

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-2ed842deeffb5058 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099093
Fingerprint: f6-e648bfe164bf7899

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:             -36,              3;
  2:            1260,           -213,               9;
  3:          -44064,          11232,            -954,            27;
  4:         1540944,        -525096,           67077,         -3807,            81;
  5:       -53887680,       22989744,        -3922020,        334449,        -14256,         243;
  6:      1884482496,     -965767536,       206171892,     -23467941,       1502226,      -51273,         729;
  7:    -65901413376,    39431784960,    -10109328768,    1439555328,    -122967396,     6300990,     -179334,     2187;
  8:   2304609511680, -1576827419904,    471911699808,  -80688616560,    8621008281,  -589385322,    25178931,  -614547,     6561;
  9: -80593491538944, 62062367152896, -21237072262272, 4238373317472, -543676655556, 46485158607, -2649238488, 97043751, -2073276, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(36-3*x)*P(n-1,x) - (36-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -36 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -36 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(36 - 3*x)*P[n - 1, x] - (36 - 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] = -36 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(36 - 3*x)*v[i - 1] - (36 - 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 (32/3, 12]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-2a73ae63953e9ff6
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-2a73ae63953e9ff6
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-2a73ae63953e9ff6

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-2a73ae63953e9ff6

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-e0b7b31567b585e1 basis_map (A038327) → 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