A113953 → x ↦ x−10

Proved for all n: all roots real, in (2, 10] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-10 + x\right)\,P_{n-1}(x) + \left(-20 + 2\,x\right)\,P_{n-2}(x)\)

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

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

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	1														-9
2	80	-18	1													63
3	-600	220	-26	1												-405
4	4400	-2280	424	-34	1											2511
5	-32000	21600	-5560	692	-42	1										-15309
6	232000	-193600	64160	-10952	1024	-50	1									92583
7	-1680000	1672000	-680800	148720	-18968	1420	-58	1								-557685
8	12160000	-14064000	6809600	-1820640	296016	-30120	1880	-66	1							3352671
9	-88000000	116000000	-65200000	20680000	-4104000	530880	-44920	2404	-74	1						-20135709
10	636800000	-942400000	603680000	-221968000	52158400	-8218368	882240	-63880	2992	-82	1					120873303
11	-4608000000	7564800000	-5443200000	2279360000	-620112000	115516480	-15080608	1383120	-87512	3644	-90	1				-725416965
12	33344000000	-60134400000	48038400000	-22590080000	6993376000	-1506592640	232241024	-25869728	2070640	-116328	4360	-98	1			4353033231
13	-241280000000	474176000000	-416524800000	217465600000	-75562880000	18508748800	-3296357760	433114688	-42059648	2986016	-150840	5140	-106	1		-26119793709
14	1745920000000	-3713664000000	3558387200000	-2043302400000	788046720000	-216531763200	43814320640	-6645628032	760558912	-65451968	4174560	-191560	5984	-114	1	156723545223

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Technical details

Node id: n-f5c69f2b67904644 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113953
Fingerprint: f6-b469fe3780d6c71d

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,         1;
  2:        80,       -18,         1;
  3:      -600,       220,       -26,        1;
  4:      4400,     -2280,       424,      -34,        1;
  5:    -32000,     21600,     -5560,      692,      -42,      1;
  6:    232000,   -193600,     64160,   -10952,     1024,    -50,      1;
  7:  -1680000,   1672000,   -680800,   148720,   -18968,   1420,    -58,    1;
  8:  12160000, -14064000,   6809600, -1820640,   296016, -30120,   1880,  -66,   1;
  9: -88000000, 116000000, -65200000, 20680000, -4104000, 530880, -44920, 2404, -74, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(10-x)*P(n-1,x) - (20-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -10 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -10 + x;
P[n_, x_] := P[n, x] = Expand[-(10 - x)*P[n - 1, x] - (20 - 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 + x; for(n = 2, N, my(i = n + 1); v[i] = -(10 - x)*v[i - 1] - (20 - 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 (2, 10]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-ba5819a43d999a58
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-ba5819a43d999a58
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-ba5819a43d999a58

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-ba5819a43d999a58

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): A Jacobsthal triangle.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
  - theorem_check ma_wang_interval e-2d5b16bf2955a407 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-1078e1fe9c978d23 basis_map (A038303) → A Jacobsthal triangle.

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