A113214 → x ↦ x−1

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	-2	1	1													0
3	1	-3	1	1												0
4	1	1	-4	1	1											0
5	-2	4	1	-5	1	1										0
6	1	-6	8	1	-6	1	1									0
7	1	1	-11	13	1	-7	1	1								0
8	-2	7	-2	-17	19	1	-8	1	1							0
9	1	-9	21	-9	-24	26	1	-9	1	1						0
10	1	1	-21	45	-21	-32	34	1	-10	1	1					0
11	-2	10	-8	-36	81	-39	-41	43	1	-11	1	1				0
12	1	-12	40	-38	-51	131	-64	-51	53	1	-12	1	1			0
13	1	1	-34	106	-104	-62	197	-97	-62	64	1	-13	1	1		0
14	-2	13	-17	-62	223	-224	-64	281	-139	-74	76	1	-14	1	1	0

Generated b-files: A130706 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-d2980b44b9e1faf3 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint: f6-6ca9b4f5705da4eb

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 ?

A130779: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
A130706: row sums, 15 terms agree OEIS nonn (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:  1,  1;
  2: -2,  1,   1;
  3:  1, -3,   1,   1;
  4:  1,  1,  -4,   1,   1;
  5: -2,  4,   1,  -5,   1,  1;
  6:  1, -6,   8,   1,  -6,  1,  1;
  7:  1,  1, -11,  13,   1, -7,  1,  1;
  8: -2,  7,  -2, -17,  19,  1, -8,  1, 1;
  9:  1, -9,  21,  -9, -24, 26,  1, -9, 1, 1;

%F Formula

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

%t Mathematica

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

by the cited steps: e-fa99d43ba99a1c05
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-fa99d43ba99a1c05

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-fa99d43ba99a1c05

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1+2x,x(1+x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
  - theorem_check ma_wang_interval e-5a8f629e61d609b9 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-719615516772b132 basis_map (A007318) → Riordan array (1+2x,x(1+x)).

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