A099040 → x ↦ x−2

Proved for all n: all roots real, in (0, 2] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-4 + 2\,x\right)\,P_{n-1}(x) + \left(-4 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -4 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-4	2							-2
2	12	-14	4						2
3	-32	64	-40	8					0
4	80	-240	244	-104	16				-4
5	-192	800	-1168	792	-256	32			8
6	448	-2464	4816	-4600	2336	-608	64		-8
7	-1024	7168	-17920	22528	-15936	6464	-1408	128	0

Show all 15 stored rows

Generated b-files: A108520 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-40f758b5452c976a · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-a4597142dfb843e9

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 ?

A108520: 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:    -4,      2;
  2:    12,    -14,       4;
  3:   -32,     64,     -40,      8;
  4:    80,   -240,     244,   -104,      16;
  5:  -192,    800,   -1168,    792,    -256,     32;
  6:   448,  -2464,    4816,  -4600,    2336,   -608,      64;
  7: -1024,   7168,  -17920,  22528,  -15936,   6464,   -1408,   128;
  8:  2304, -19968,   61824, -97920,   90256, -50624,   17088, -3200,   256;
  9: -5120,  53760, -201216, 389376, -448064, 325280, -151040, 43648, -7168, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(4-2*x)*P(n-1,x) - (4-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -4 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -4 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(4 - 2*x)*P[n - 1, x] - (4 - 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] = -4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(4 - 2*x)*v[i - 1] - (4 - 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 (0, 2]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-8c4e4f825f4ab306
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-8c4e4f825f4ab306
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-8c4e4f825f4ab306

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-8c4e4f825f4ab306

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-72107609b2d5e6bc basis_map (A038207) → 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