A063967 → x ↦ x−7

Proved for all n: all roots real, in (2, 6] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

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

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

\(P_{2}(x) = -6 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	-6	1														-5
3	30	-11	1													20
4	-144	84	-16	1												-75
5	684	-552	163	-21	1											275
6	-3240	3348	-1350	267	-26	1										-1000
7	15336	-19332	9918	-2663	396	-31	1									3625
8	-72576	108000	-67392	22944	-4616	550	-36	1								-13125
9	343440	-589248	433512	-179160	45601	-7334	729	-41	1							47500
10	-1625184	3158352	-2677968	1303416	-402126	81689	-10942	933	-46	1						-171875
11	7690464	-16696368	16035840	-8989992	3263406	-802655	135633	-15565	1162	-51	1					621875
12	-36391680	87293376	-93685248	59477328	-24854256	7187076	-1469112	212483	-21328	1416	-56	1				-2250000
13	172207296	-452283264	536493456	-380573424	180032436	-59897376	14385295	-2514987	317914	-28356	1695	-61	1			8140625
14	-814893696	2325754944	-3021839136	2369384784	-1252165176	471439980	-130207398	26731207	-4082020	458226	-36774	1999	-66	1		-29453125
15	3856118400	-11883516480	16785545760	-14418213840	8421607800	-3541388364	1106475222	-261119423	46800856	-6343326	640344	-46707	2328	-71	1	106562500

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Technical details

Node id: n-50a9a09f321fcea8 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A063967
Fingerprint: f6-4c21f592c13b2a9c

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:       -6,       1;
   3:       30,     -11,        1;
   4:     -144,      84,      -16,       1;
   5:      684,    -552,      163,     -21,       1;
   6:    -3240,    3348,    -1350,     267,     -26,     1;
   7:    15336,  -19332,     9918,   -2663,     396,   -31,      1;
   8:   -72576,  108000,   -67392,   22944,   -4616,   550,    -36,   1;
   9:   343440, -589248,   433512, -179160,   45601, -7334,    729, -41,   1;
  10: -1625184, 3158352, -2677968, 1303416, -402126, 81689, -10942, 933, -46, 1;

%F Formula

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

%t Mathematica

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

by the cited steps: e-12ad66072da87843
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-12ad66072da87843
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-12ad66072da87843

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-12ad66072da87843

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-07199d0e633dc35b basis_map (A027466) → 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