A113953 → x ↦ x−9

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

Recurrence

\(P_{n}(x) = \left(-9 + x\right)\,P_{n-1}(x) + \left(-18 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -9 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-9	1														-8
2	63	-16	1													48
3	-405	171	-23	1												-256
4	2511	-1530	328	-30	1											1280
5	-15309	12393	-3726	534	-37	1										-6144
6	92583	-94284	36963	-7336	789	-44	1									28672
7	-557685	687447	-335097	85923	-12703	1093	-51	1								-131072
8	3352671	-4862430	2849418	-902430	171376	-20170	1446	-58	1							589824
9	-20135709	33625125	-23100552	8754480	-2044314	307826	-30080	1848	-65	1						-2621440
10	120873303	-228532752	180515709	-79948296	22263678	-4108936	512178	-42776	2299	-72	1					11534336
11	-725416965	1532144403	-1370113947	696268629	-226014786	49614606	-7561446	803738	-58601	2799	-79	1				-50331648
12	4353033231	-10159380450	10156821660	-5836430862	2169758907	-554058036	100230544	-13000764	1204213	-77898	3348	-86	1			218103808
13	-26119793709	66758024097	-73844435538	47411636202	-19903457619	5811188751	-1220797692	187647244	-21176387	1737711	-101010	3946	-93	1		-939524096
14	156723545223	-435367096020	528216393159	-375179762520	175814232723	-57891593916	13886102115	-2475148048	330557365	-33005196	2430741	-128280	4593	-100	1	4026531840

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Technical details

Node id: n-a8f637a6283d3e7d · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113953
Fingerprint: f6-fdd9049b0d28e334

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:        -9,        1;
  2:        63,      -16,         1;
  3:      -405,      171,       -23,       1;
  4:      2511,    -1530,       328,     -30,        1;
  5:    -15309,    12393,     -3726,     534,      -37,      1;
  6:     92583,   -94284,     36963,   -7336,      789,    -44,      1;
  7:   -557685,   687447,   -335097,   85923,   -12703,   1093,    -51,    1;
  8:   3352671, -4862430,   2849418, -902430,   171376, -20170,   1446,  -58,   1;
  9: -20135709, 33625125, -23100552, 8754480, -2044314, 307826, -30080, 1848, -65, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(9-x)*P(n-1,x) - (18-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -9 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -9 + x;
P[n_, x_] := P[n, x] = Expand[-(9 - x)*P[n - 1, x] - (18 - 2*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] = -9 + x; for(n = 2, N, my(i = n + 1); v[i] = -(9 - x)*v[i - 1] - (18 - 2*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 (1, 9]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-f1a9ca5f4baaa1dd
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-f1a9ca5f4baaa1dd
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-f1a9ca5f4baaa1dd

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-f1a9ca5f4baaa1dd

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): A Jacobsthal triangle.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
  - theorem_check ma_wang_interval e-2d5b16bf2955a407 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-a2d6a4286b164d3d basis_map (A038291) → A Jacobsthal triangle.

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