A099093 → x ↦ x−5

Proved for all n: all roots real, in (11/3, 5] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

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

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

\(P_{1}(x) = -15 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-15	3							-12
2	210	-87	9						132
3	-2925	1845	-387	27					-1440
4	40725	-34515	10944	-1539	81				15696
5	-567000	603450	-256365	54351	-5751	243			-171072
6	7894125	-10112850	5388120	-1528443	243486	-20655	729		1864512
7	-109906875	164622375	-105504525	37506645	-7988301	1019385	-72171	2187	-20321280

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-ce49db683d596495 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099093
Fingerprint: f6-38dc195ff67e56ef

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:          -15,           3;
  2:          210,         -87,            9;
  3:        -2925,        1845,         -387,          27;
  4:        40725,      -34515,        10944,       -1539,          81;
  5:      -567000,      603450,      -256365,       54351,       -5751,        243;
  6:      7894125,   -10112850,      5388120,    -1528443,      243486,     -20655,        729;
  7:   -109906875,   164622375,   -105504525,    37506645,    -7988301,    1019385,     -72171,     2187;
  8:   1530191250, -2623681125,   1965274650,  -840022245,   224106831,  -38215395,    4067820,  -247131,    6561;
  9: -21304265625, 41146734375, -35273728125, 17617044375, -5649324750, 1206295740, -171522765, 15661107, -833247, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(15-3*x)*P(n-1,x) - (15-3*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -15 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -15 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(15 - 3*x)*P[n - 1, x] - (15 - 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] = -15 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(15 - 3*x)*v[i - 1] - (15 - 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 (11/3, 5]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-ae28d4484158822f
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-ae28d4484158822f
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-ae28d4484158822f

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-ae28d4484158822f

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-089b35b0d9fc7c83 basis_map (A038243) → 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