A099097 → x ↦ x−10

Proved for all n: all roots real, in (86/9, 10] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-30 + 3\,x\right)\,P_{n-1}(x) + \left(-10 + x\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = -30 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-30	3							-27
3	890	-179	9						720
4	-26400	7980	-804	27					-19197
5	783100	-315920	47791	-3213	81				511839
6	-23229000	11720700	-2365470	238689	-12042	243			-13646880
7	689039000	-417365700	105332370	-14177159	1073304	-43335	729		363859209
8	-20438880000	14447652000	-4376692800	736559520	-74371488	4505490	-151632	2187	-9701376723

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-1b374fdc6d596a8d · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint: f6-9427854bce2c69b6

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:             -30,              3;
   3:             890,           -179,              9;
   4:          -26400,           7980,           -804,            27;
   5:          783100,        -315920,          47791,         -3213,            81;
   6:       -23229000,       11720700,       -2365470,        238689,        -12042,         243;
   7:       689039000,     -417365700,      105332370,     -14177159,       1073304,      -43335,         729;
   8:    -20438880000,    14447652000,    -4376692800,     736559520,     -74371488,     4505490,     -151632,     2187;
   9:    606276010000,  -489883504000,   173173050600,  -34979760040,    4415913001,  -356772510,    18014805,  -519777,     6561;
  10: -17983891500000, 16350417750000, -6606627450000, 1557169665000, -235936395750, 23831487915, -1604739870, 69464223, -1753974, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(30-3*x)*P(n-1,x) - (10-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -30 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -30 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(30 - 3*x)*P[n - 1, x] - (10 - 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] = -30 + 3*x; for(n = 3, N, my(i = n); v[i] = -(30 - 3*x)*v[i - 1] - (10 - 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 (86/9, 10]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-c52c3082cda5f9b6
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-c52c3082cda5f9b6
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-c52c3082cda5f9b6

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-c52c3082cda5f9b6

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-85ae0250e4103468 basis_map (A038303) → 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