A099097 → x ↦ x−6

Proved for all n: all roots real, in (50/9, 6] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-18 + 3\,x\right)\,P_{n-1}(x) + \left(-6 + x\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = -18 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	-18	3														-15
3	318	-107	9													220
4	-5616	2844	-480	27												-3225
5	99180	-67080	17011	-1917	81											47275
6	-1751544	1482300	-501714	84897	-7182	243										-693000
7	30932712	-31434372	13308606	-3004775	381564	-25839	729									10158625
8	-546279552	647971488	-329365440	93000672	-15754488	1601154	-90396	2187								-148914375
9	9647435664	-13082786496	7761206376	-2630771160	557288641	-75547638	6400377	-309825	6561							2182922500
10	-170376164640	259998355440	-176325910128	69750130536	-17735981418	3006361995	-339706170	24674463	-1045386	19683						-31999265625
11	3008886349536	-5102954737200	3894211423872	-1760934246696	522523554126	-106311885695	15019847145	-1454999517	92491146	-3483891	59049					469074371875
12	-53137697303808	99149478020928	-84086716025088	42784624018800	-14512060695312	3445410451500	-584248306680	70761779793	-5998892400	339019992	-11494872	177147				-6876119250000
13	938425233371328	-1910477081515392	1782541099233936	-1007923563509616	384674889000564	-104393175344640	20656269906175	-3002707112067	318255456186	-23985642564	1220129487	-37614213	531441			100796416890625
14	-16572828016861056	36555828601961664	-37213501256585568	23149452980737008	-9818061704348328	2997917299800900	-678041484053490	115008718750293	-14629964413356	1378474922358	-93511269234	4324886541	-122231430	1594323		-1477565657109375
15	292680352903271040	-695322111163429440	766904784747506208	-520500574942731216	242857496722967928	-82405662568392780	20970167817583830	-4065608877087167	603453275842104	-68240272530942	5787321193800	-356935950675	15134022504	-394860663	4782969	21659502772187500

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Technical details

Node id: n-f92327b2555fefca · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint: f6-bdddce713470c1d7

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:           -18,            3;
   3:           318,         -107,             9;
   4:         -5616,         2844,          -480,          27;
   5:         99180,       -67080,         17011,       -1917,           81;
   6:      -1751544,      1482300,       -501714,       84897,        -7182,        243;
   7:      30932712,    -31434372,      13308606,    -3004775,       381564,     -25839,        729;
   8:    -546279552,    647971488,    -329365440,    93000672,    -15754488,    1601154,     -90396,     2187;
   9:    9647435664, -13082786496,    7761206376, -2630771160,    557288641,  -75547638,    6400377,  -309825,     6561;
  10: -170376164640, 259998355440, -176325910128, 69750130536, -17735981418, 3006361995, -339706170, 24674463, -1045386, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(18-3*x)*P(n-1,x) - (6-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -18 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -18 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(18 - 3*x)*P[n - 1, x] - (6 - 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] = -18 + 3*x; for(n = 3, N, my(i = n); v[i] = -(18 - 3*x)*v[i - 1] - (6 - 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 (50/9, 6]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-b062da8ba0c30336
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-b062da8ba0c30336
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-b062da8ba0c30336

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-b062da8ba0c30336

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-b69b8b071f02daa4 basis_map (A038255) → 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