A099040 → x ↦ x−10

Proved for all n: all roots real, in (8, 10] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

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

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

\(P_{1}(x) = -20 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-20	2														-18
2	380	-78	4													306
3	-7200	2240	-232	8												-5184
4	136400	-56880	8884	-616	16											87804
5	-2584000	1351200	-282320	29464	-1536	32										-1487160
6	48952000	-30781600	8057360	-1123832	88096	-3680	64									25188408
7	-927360000	681344000	-214361600	37437440	-3919936	246080	-8576	128								-426622464
8	17568160000	-14768064000	5427209600	-1138880640	149264016	-12511680	655040	-19584	256							7225813008
9	-332816000000	315016000000	-132430400000	32454560000	-5109768000	536000160	-37460480	1682048	-44032	512						-122385429792
10	6304956800000	-6635454400000	3140559680000	-880319968000	161841438400	-20390777568	1783085760	-106860160	4200448	-97792	1024					2072873102112
11	-119442816000000	138353049600000	-72803462400000	22973566720000	-4830364224000	710558888960	-74622060416	5594812800	-293484544	10258432	-215040	2048				-35108778101760
12	2262757184000000	-2860627622400000	1656693244800000	-581190740480000	137556949216000	-23140407799040	2837115715904	-255437002112	16761587200	-781780992	24601600	-468992	4096			594646289993664
13	-42866287360000000	58732120192000000	-37122344793600000	14332123040000000	-3770966047360000	714050148185600	-100109570929920	10521831097216	-829046431744	48366656512	-2030776320	58112000	-1015808	8192		-10071675214054272
14	812070603520000000	-1198636911744000000	821056016115200000	-345949949088000000	100170046561920000	-21085013004019200	3327268585053440	-399872792330112	36778485081088	-2576267199488	135293245440	-5165209600	135520256	-2187264	16384	170586520633090944

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Technical details

Node id: n-dd0db9eabdf2db44 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-fc444432470a0fef

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:           -20,            2;
  2:           380,          -78,             4;
  3:         -7200,         2240,          -232,           8;
  4:        136400,       -56880,          8884,        -616,          16;
  5:      -2584000,      1351200,       -282320,       29464,       -1536,        32;
  6:      48952000,    -30781600,       8057360,    -1123832,       88096,     -3680,        64;
  7:    -927360000,    681344000,    -214361600,    37437440,    -3919936,    246080,     -8576,     128;
  8:   17568160000, -14768064000,    5427209600, -1138880640,   149264016, -12511680,    655040,  -19584,    256;
  9: -332816000000, 315016000000, -132430400000, 32454560000, -5109768000, 536000160, -37460480, 1682048, -44032, 512;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -20 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(20 - 2*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] = -20 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(20 - 2*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 (8, 10]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-c2f106499058af8f
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-c2f106499058af8f
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-c2f106499058af8f

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-c2f106499058af8f

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-532ccda39f04a0f0 basis_map (A038303) → 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