A099097 → x ↦ x−5

Proved for all n: all roots real, in (41/9, 5] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-15 + 3\,x\right)\,P_{n-1}(x) + \left(-5 + x\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = -15 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-15	3							-12
3	220	-89	9						140
4	-3225	1965	-399	27					-1632
5	47275	-38485	11746	-1593	81				19024
6	-693000	706050	-287685	58599	-5967	243			-221760
7	10158625	-12430050	6336210	-1722329	263304	-21465	729		2585024
8	-148914375	212703375	-130188825	44262885	-9028113	1104705	-75087	2187	-30133248

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-936b9851d7f61099 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint: f6-14f92c27ff4cd655

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:          -15,           3;
   3:          220,         -89,            9;
   4:        -3225,        1965,         -399,          27;
   5:        47275,      -38485,        11746,       -1593,          81;
   6:      -693000,      706050,      -287685,       58599,       -5967,        243;
   7:     10158625,   -12430050,      6336210,    -1722329,      263304,     -21465,        729;
   8:   -148914375,   212703375,   -130188825,    44262885,    -9028113,    1104705,     -75087,     2187;
   9:   2182922500, -3564984875,   2546831400, -1039561895,   265171501,  -43284285,    4415310,  -257337,    6561;
  10: -31999265625, 58811109375, -48033778125, 22882419375, -7006854750, 1430227140, -194602365, 17019963, -868239, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(15-3*x)*P(n-1,x) - (5-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -15 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -15 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(15 - 3*x)*P[n - 1, x] - (5 - 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] = -15 + 3*x; for(n = 3, N, my(i = n); v[i] = -(15 - 3*x)*v[i - 1] - (5 - 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 (41/9, 5]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-4899b779b1a74e41
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-4899b779b1a74e41
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-4899b779b1a74e41

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-4899b779b1a74e41

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-bcf40186578fecd4 basis_map (A038243) → 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