A063967 → x ↦ x−12

Proved for all n: all roots real, in (7, 11] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-11 + x\right)\,P_{n-1}(x) + \left(-11 + x\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = -11 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	-11	1														-10
3	110	-21	1													90
4	-1089	319	-31	1												-800
5	10769	-4257	628	-41	1											7100
6	-106480	52998	-10505	1037	-51	1										-63000
7	1052821	-631862	157388	-20833	1546	-61	1									559000
8	-10409751	7313845	-2194577	364639	-36241	2155	-71	1								-4960000
9	102926230	-82858743	29091062	-5819055	725451	-57729	2864	-81	1							44010000
10	-1017681269	923510357	-371406233	86895061	-13035726	1300524	-86297	3673	-91	1						-390500000
11	10062305429	-10161922793	4606118495	-1234151237	216489031	-25981020	2160558	-122945	4582	-101	1					3464900000
12	-99490865760	110667160956	-55820247318	16854530198	-3385242531	474938761	-47497367	3386253	-168673	5591	-111	1				-30744000000
13	983714163641	-1194986180124	663860655216	-223038297394	50476667461	-8107288651	947662640	-81233197	5068309	-224481	6700	-121	1			272791000000
14	-9726456276691	12811732508729	-7772763506046	2876061847054	-724189441426	131047273720	-17534167893	1756481657	-131742940	7307426	-291369	7909	-131	1		-2420470000000
15	96170163243550	-136526951727705	89814677687735	-36292161897090	10063864063275	-2026052609724	305391542852	-35014238313	3068669401	-204587026	10214304	-370337	9218	-141	1	21476790000000

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Technical details

Node id: n-383c73dff2796e24 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A063967
Fingerprint: f6-b02960c001dcda1c

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:         -11,         1;
   3:         110,       -21,          1;
   4:       -1089,       319,        -31,        1;
   5:       10769,     -4257,        628,      -41,         1;
   6:     -106480,     52998,     -10505,     1037,       -51,       1;
   7:     1052821,   -631862,     157388,   -20833,      1546,     -61,      1;
   8:   -10409751,   7313845,   -2194577,   364639,    -36241,    2155,    -71,    1;
   9:   102926230, -82858743,   29091062, -5819055,    725451,  -57729,   2864,  -81,   1;
  10: -1017681269, 923510357, -371406233, 86895061, -13035726, 1300524, -86297, 3673, -91, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(11-x)*P(n-1,x) - (11-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -11 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -11 + x;
P[n_, x_] := P[n, x] = Expand[-(11 - x)*P[n - 1, x] - (11 - 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] = -11 + x; for(n = 3, N, my(i = n); v[i] = -(11 - x)*v[i - 1] - (11 - 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 (7, 11]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-4d81c1e2745a9ed0
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-4d81c1e2745a9ed0
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-4d81c1e2745a9ed0

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-4d81c1e2745a9ed0

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-5, -1]
  - theorem_check ma_wang_interval e-32a5b77539bbfdf5 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-37edf5cceba94f47 basis_map (A038327) → Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) …

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