A099097 → x ↦ x−1

Proved for all n: all roots real, in (5/9, 1] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	-3	3														0
3	8	-17	9													0
4	-21	69	-75	27												0
5	55	-245	406	-297	81											0
6	-144	810	-1809	2007	-1107	243										0
7	377	-2562	7206	-10745	8964	-3969	729									0
8	-987	7863	-26685	50037	-56013	37449	-13851	2187								0
9	2584	-23611	93876	-212215	298441	-267453	149202	-47385	6561							0
10	-6765	69735	-317913	841551	-1425918	1604220	-1198665	573723	-159651	19683						0
11	17711	-203305	1045457	-3172301	6291751	-8524520	7992000	-5120577	2146176	-531441	59049					0
12	-46368	586548	-3358638	11493810	-26124687	41418675	-46746675	37565343	-21066885	7853517	-1751787	177147				0
13	121393	-1677732	10586796	-40339586	103391439	-187813815	247979530	-239823477	168629931	-84083589	28225422	-5727753	531441			0
14	-317811	4764459	-32848398	137926698	-393574578	806072400	-1219214685	1379097003	-1166727996	729220158	-327321729	99930591	-18600435	1594323		0
15	832040	-13447685	100574043	-461398906	1450772803	-3307735680	5640067910	-7307132057	7229021589	-5435130942	3057316650	-1247803785	349333884	-60052833	4782969	0

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Technical details

Node id: n-b0cf60708a253825 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint: f6-20b7e8ee86b20d50

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:    -3,      3;
   3:     8,    -17,       9;
   4:   -21,     69,     -75,      27;
   5:    55,   -245,     406,    -297,       81;
   6:  -144,    810,   -1809,    2007,    -1107,     243;
   7:   377,  -2562,    7206,  -10745,     8964,   -3969,      729;
   8:  -987,   7863,  -26685,   50037,   -56013,   37449,   -13851,   2187;
   9:  2584, -23611,   93876, -212215,   298441, -267453,   149202, -47385,    6561;
  10: -6765,  69735, -317913,  841551, -1425918, 1604220, -1198665, 573723, -159651, 19683;

%F Formula

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

%t Mathematica

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

by the cited steps: e-5d14c25d8e006de4
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-5d14c25d8e006de4
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-5d14c25d8e006de4

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-5d14c25d8e006de4

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-cb03f5f41d038dd3 basis_map (A007318) → 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