A030528 → x ↦ x−3

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

Recurrence

\(P_{n}(x) = \left(-3 + x\right)\,P_{n-1}(x) + \left(-3 + x\right)\,P_{n-2}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-2	1														-1
2	3	-4	1													0
3	-3	10	-6	1												2
4	0	-18	21	-8	1											-4
5	9	21	-53	36	-10	1										4
6	-27	0	99	-116	55	-12	1									0
7	54	-81	-117	286	-215	78	-14	1								-8
8	-81	270	-27	-528	650	-358	105	-16	1							16
9	81	-594	621	582	-1547	1275	-553	136	-18	1						-16
10	0	972	-2106	432	2745	-3648	2261	-808	171	-20	1					0
11	-243	-1053	4833	-4527	-2580	8317	-7497	3724	-1131	210	-22	1				32
12	729	0	-8262	15012	-4590	-13842	20377	-13984	5796	-1530	253	-24	1			-64
13	-1458	3645	9234	-34884	31995	9405	-44165	43660	-24255	8625	-2013	300	-26	1		64
14	2187	-11664	729	60588	-102087	40716	66927	-112816	85053	-39744	12375	-2588	351	-28	1	0

Generated b-files: A009116 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-81eeb524d0837717 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A030528
Fingerprint: f6-9f0f04d1052f691c

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 ?

A009116: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:  -2,    1;
  2:   3,   -4,    1;
  3:  -3,   10,   -6,    1;
  4:   0,  -18,   21,   -8,     1;
  5:   9,   21,  -53,   36,   -10,    1;
  6: -27,    0,   99, -116,    55,  -12,    1;
  7:  54,  -81, -117,  286,  -215,   78,  -14,   1;
  8: -81,  270,  -27, -528,   650, -358,  105, -16,   1;
  9:  81, -594,  621,  582, -1547, 1275, -553, 136, -18, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(3-x)*P(n-1,x) - (3-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -2 + x;
P[n_, x_] := P[n, x] = Expand[-(3 - x)*P[n - 1, x] - (3 - 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] = -2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(3 - x)*v[i - 1] - (3 - 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-a2e7017ebd9e3749
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-a2e7017ebd9e3749

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-a2e7017ebd9e3749

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
  - theorem_check ma_wang_interval e-f811a014585c6352 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-5c2b8bc6b43c730e basis_map (A027465) → Triangle read by rows: T(n,k) = binomial(k,n-k).

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