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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	1530191250	-2623681125	1965274650	-840022245	224106831	-38215395	4067820	-247131	6561							221481216
9	-21304265625	41146734375	-35273728125	17617044375	-5649324750	1206295740	-171522765	15661107	-833247	19683						-2413919232
10	296611115625	-637168021875	615195961875	-351580692375	131709335175	-33796858932	6016065210	-733574475	58642218	-2775303	59049					26309256192
11	-4129602750000	9766239862500	-10486897368750	6749221421250	-2892791100375	867038479155	-185439826251	28302327855	-3020874669	214761213	-9152595	177147				-286744043520
12	57494874515625	-148435052512500	175462736625000	-125579715153750	60609148664625	-20781869598945	5191081276284	-951802583823	127139746905	-12066486003	772360920	-29937843	531441			3125217447936
13	-800479076484375	2240128005046875	-2890644026793750	2277384923756250	-1222236844661250	471871539489600	-134829115109865	28869428189619	-4632283851444	550132488558	-47003299245	2736035415	-97253703	1594323		-34061680852992
14	11144763029531250	-33604346893921875	47002798210134375	-40422621999543750	23879831065756875	-10251228136349700	3297839517175680	-807678485587683	151330038385473	-21586422351942	2307662082540	-179284278555	9569126646	-314081631	4782969	371237560860672

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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