A099040 → x ↦ x−3

Proved for all n: all roots real, in (1, 3] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-6	2														-4
2	30	-22	4													12
3	-144	168	-64	8												-32
4	684	-1104	652	-168	16											80
5	-3240	6696	-5400	2136	-416	32										-192
6	15336	-38664	39672	-21304	6336	-992	64									448
7	-72576	216000	-269568	183552	-73856	17600	-2304	128								-1024
8	343440	-1178496	1734048	-1433280	729616	-234688	46656	-5248	256							2304
9	-1625184	6316704	-10711872	10427328	-6434016	2614048	-700288	119424	-11776	512						-5120
10	7690464	-33392736	64143360	-71919936	52214496	-25684960	8680512	-1992320	297472	-26112	1024					11264
11	-36391680	174586752	-374740992	475818624	-397668096	229986432	-94023168	27197824	-5459968	724992	-57344	2048				-24576
12	172207296	-904566528	2145973824	-3044587392	2880518976	-1916716032	920658880	-321918336	81385984	-14518272	1735680	-124928	4096			53248
13	-814893696	4651509888	-12087356544	18955078272	-20034642816	15086079360	-8333273472	3421594496	-1044997120	234611712	-37656576	4093952	-270336	8192		-114688
14	3856118400	-23767032960	67142183040	-115345710720	134745724800	-113324427648	70814414208	-33423286144	11981019136	-3247782912	655712256	-95655936	9535488	-581632	16384	245760

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Technical details

Node id: n-b59b1769d0dc40fc · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-d773d6147703cd7e

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:       -6,        2;
  2:       30,      -22,         4;
  3:     -144,      168,       -64,        8;
  4:      684,    -1104,       652,     -168,       16;
  5:    -3240,     6696,     -5400,     2136,     -416,      32;
  6:    15336,   -38664,     39672,   -21304,     6336,    -992,      64;
  7:   -72576,   216000,   -269568,   183552,   -73856,   17600,   -2304,    128;
  8:   343440, -1178496,   1734048, -1433280,   729616, -234688,   46656,  -5248,    256;
  9: -1625184,  6316704, -10711872, 10427328, -6434016, 2614048, -700288, 119424, -11776, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(6-2*x)*P(n-1,x) - (6-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -6 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -6 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(6 - 2*x)*P[n - 1, x] - (6 - 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] = -6 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(6 - 2*x)*v[i - 1] - (6 - 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, 3]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-319b2a76ea6c1ab0
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-319b2a76ea6c1ab0
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-319b2a76ea6c1ab0

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-319b2a76ea6c1ab0

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1, 2+2x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
  - theorem_check ma_wang_interval e-17786677d2ed021b root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-af79ef336fd9f8da basis_map (A027465) → Riordan array (1, 2+2x).

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