A063967 → x ↦ x−9

Proved for all n: all roots real, in (4, 8] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-8 + x\right)\,P_{n-1}(x) + \left(-8 + x\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = -8 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	-8	1														-7
3	56	-15	1													42
4	-384	160	-22	1												-245
5	2624	-1488	313	-29	1											1421
6	-17920	12864	-3656	515	-36	1										-8232
7	122368	-106304	38120	-7231	766	-43	1									47677
8	-835584	851968	-369152	88192	-12556	1066	-50	1								-276115
9	5705728	-6678528	3393920	-978720	175281	-19974	1415	-57	1							1599066
10	-38961152	51482624	-30024704	10148992	-2192328	313989	-29828	1813	-64	1						-9260657
11	266043392	-391688192	257850368	-99992960	25306648	-4369167	521319	-42461	2260	-71	1					53631137
12	-1816657920	2948726784	-2162810880	946577408	-274758528	55555744	-7987106	816675	-58216	2756	-78	1				-310593360
13	12404916224	-22006923264	17796722688	-8677636096	2842199488	-658944496	110912873	-13659499	1221862	-77436	3301	-85	1			1798735561
14	-84706066432	163053830144	-144129490944	77482381312	-28270586368	7394550976	-1426794888	205668359	-22151992	1761086	-100464	3895	-92	1		-10416995407
15	578409201664	-1200676405248	1151709052928	-676770729984	272231840256	-79313238720	17262662600	-2851952895	359449900	-34399330	2460954	-127643	4538	-99	1	60327818922

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Technical details

Node id: n-29ef532ce8637fba · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A063967
Fingerprint: f6-185ebee0b00e059f

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:        -8,        1;
   3:        56,      -15,         1;
   4:      -384,      160,       -22,        1;
   5:      2624,    -1488,       313,      -29,        1;
   6:    -17920,    12864,     -3656,      515,      -36,      1;
   7:    122368,  -106304,     38120,    -7231,      766,    -43,      1;
   8:   -835584,   851968,   -369152,    88192,   -12556,   1066,    -50,    1;
   9:   5705728, -6678528,   3393920,  -978720,   175281, -19974,   1415,  -57,   1;
  10: -38961152, 51482624, -30024704, 10148992, -2192328, 313989, -29828, 1813, -64, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(8-x)*P(n-1,x) - (8-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -8 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -8 + x;
P[n_, x_] := P[n, x] = Expand[-(8 - x)*P[n - 1, x] - (8 - 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] = -8 + x; for(n = 3, N, my(i = n); v[i] = -(8 - x)*v[i - 1] - (8 - 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 (4, 8]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-61f403043a781326
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-61f403043a781326
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-61f403043a781326

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-61f403043a781326

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-19f9843f81ec4d82 basis_map (A038291) → 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