A099040 → x ↦ x−5

Proved for all n: all roots real, in (3, 5] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-10 + 2\,x\right)\,P_{n-1}(x) + \left(-10 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -10 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-10	2														-8
2	90	-38	4													56
3	-800	520	-112	8												-384
4	7100	-6240	2044	-296	16											2624
5	-63000	69800	-30760	6744	-736	32										-17920
6	559000	-747400	414280	-121912	20096	-1760	64									122368
7	-4960000	7768000	-5190400	1918720	-423936	56000	-4096	128								-835584
8	44010000	-79008000	61802400	-27520320	7632016	-1350080	148800	-9344	256							5705728
9	-390500000	790500000	-708600000	369240000	-123284000	27356960	-4035200	381568	-20992	512						-38961152
10	3464900000	-7807900000	7890960000	-4710792000	1839959200	-491372768	90877760	-11495040	951808	-46592	1024					266043392
11	-30744000000	76322800000	-85858400000	57780240000	-25849856000	8073508480	-1796457216	284819840	-31535104	2322432	-102400	2048				-1816657920
12	272791000000	-739707200000	916704200000	-686629360000	346237864000	-123841150720	32220065984	-6144406912	852482560	-83924992	5565440	-223232	4096			12404916224
13	-2420470000000	7117938000000	-9635226800000	7950182800000	-4461578320000	1798452438400	-535771372160	119443088256	-19928648704	2457920512	-217835520	13137920	-483328	8192		-84706066432
14	21476790000000	-68077666000000	99941687600000	-90072579600000	55680511440000	-24976793788800	8384735637120	-2140089425792	417359024128	-61892287488	6870691840	-553687040	30621696	-1040384	16384	578409201664

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Technical details

Node id: n-af83e10a056d6f29 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-783d3f3b454309c7

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:        -10,         2;
  2:         90,       -38,          4;
  3:       -800,       520,       -112,         8;
  4:       7100,     -6240,       2044,      -296,         16;
  5:     -63000,     69800,     -30760,      6744,       -736,       32;
  6:     559000,   -747400,     414280,   -121912,      20096,    -1760,       64;
  7:   -4960000,   7768000,   -5190400,   1918720,    -423936,    56000,    -4096,    128;
  8:   44010000, -79008000,   61802400, -27520320,    7632016, -1350080,   148800,  -9344,    256;
  9: -390500000, 790500000, -708600000, 369240000, -123284000, 27356960, -4035200, 381568, -20992, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(10-2*x)*P(n-1,x) - (10-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -10 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -10 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(10 - 2*x)*P[n - 1, x] - (10 - 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] = -10 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(10 - 2*x)*v[i - 1] - (10 - 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 (3, 5]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-bdc3e5aca0457f81
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-bdc3e5aca0457f81
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-bdc3e5aca0457f81

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-bdc3e5aca0457f81

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-763de8a298cf812d basis_map (A038243) → 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