A099093 → x ↦ x−10

Proved for all n: all roots real, in (26/3, 10] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

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

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

\(P_{1}(x) = -30 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-30	3							-27
2	870	-177	9						702
3	-25200	7740	-792	27					-18225
4	729900	-299880	46179	-3159	81				473121
5	-21141000	10878300	-2238030	230121	-11826	243			-12282192
6	612333000	-378585900	97490790	-13384413	1033236	-42525	729		318844917
7	-17735760000	12804804000	-3960705600	680387040	-70105176	4332690	-148716	2187	-8277193575

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-e2dcd39fb6d71722 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099093
Fingerprint: f6-c7b2b5fef74944cb

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:             -30,              3;
  2:             870,           -177,              9;
  3:          -25200,           7740,           -792,            27;
  4:          729900,        -299880,          46179,         -3159,            81;
  5:       -21141000,       10878300,       -2238030,        230121,        -11826,         243;
  6:       612333000,     -378585900,       97490790,     -13384413,       1033236,      -42525,         729;
  7:    -17735760000,    12804804000,    -3960705600,     680387040,     -70105176,     4332690,     -148716,     2187;
  8:    513702810000,  -424156824000,   153175098600,  -31599723240,    4073166081,  -335920770,    17310105,  -509571,     6561;
  9: -14879011500000, 13828461750000, -5710487850000, 1375223265000, -212849835750, 21956825115, -1509605910, 66705687, -1718982, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(30-3*x)*P(n-1,x) - (30-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -30 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -30 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(30 - 3*x)*P[n - 1, x] - (30 - 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] = -30 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(30 - 3*x)*v[i - 1] - (30 - 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 (26/3, 10]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-d18687645e4e9661
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-d18687645e4e9661
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-d18687645e4e9661

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-d18687645e4e9661

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-a4925a412f743bbb basis_map (A038303) → 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