A099040 → x ↦ x−9

Proved for all n: all roots real, in (7, 9] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-18 + 2\,x\right)\,P_{n-1}(x) + \left(-18 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -18 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-18	2														-16
2	306	-70	4													240
3	-5184	1800	-208	8												-3584
4	87804	-40896	7132	-552	16											53504
5	-1487160	868968	-202824	23640	-1376	32										-798720
6	25188408	-17704008	5178600	-806968	70656	-3296	64									11923456
7	-426622464	350433216	-123234048	24051456	-2813696	197312	-7680	128								-177995776
8	7225813008	-6791993856	2790456480	-654511680	95863696	-8978368	525120	-17536	256							2657157120
9	-122385429792	129546472608	-60893125056	16682728896	-2935820448	344159008	-26876032	1348224	-39424	512						-39666581504
10	2072873102112	-2439899851104	1291356991872	-404713247040	83175655968	-11713165024	1144677696	-76654208	3366400	-87552	1024					592150790144
11	-35108778101760	45487336157568	-26769056359680	9445477060224	-2220378075648	365121779328	-42858441984	3591111040	-210497536	8220672	-192512	2048				-8839747338240
12	594646289993664	-840925663515648	544693461233472	-213689147372928	56551171180608	-10635759896832	1457664985792	-146687751552	10757274112	-560658432	19713024	-419840	4096			131961544769536
13	-10071675214054272	15436964916229248	-10913515942444416	4712234875376256	-1386441616514688	293533072324992	-46007794023552	5405352616832	-476035259392	31037432832	-1456244736	46561280	-909312	8192		-1969948758900736
14	170586520633090944	-281682764396966016	215830883167224192	-101711468066481792	32935119472020096	-7751932514375040	1367696947536000	-183756225650560	18892333465600	-1479137909760	86811119616	-3703609344	108576768	-1957888	16384	29407795426099200

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Technical details

Node id: n-192a0153bc9ffe91 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-e57604ddfb78363b

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:           -18,            2;
  2:           306,          -70,            4;
  3:         -5184,         1800,         -208,           8;
  4:         87804,       -40896,         7132,        -552,          16;
  5:      -1487160,       868968,      -202824,       23640,       -1376,        32;
  6:      25188408,    -17704008,      5178600,     -806968,       70656,     -3296,        64;
  7:    -426622464,    350433216,   -123234048,    24051456,    -2813696,    197312,     -7680,     128;
  8:    7225813008,  -6791993856,   2790456480,  -654511680,    95863696,  -8978368,    525120,  -17536,    256;
  9: -122385429792, 129546472608, -60893125056, 16682728896, -2935820448, 344159008, -26876032, 1348224, -39424, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(18-2*x)*P(n-1,x) - (18-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -18 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -18 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(18 - 2*x)*P[n - 1, x] - (18 - 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] = -18 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(18 - 2*x)*v[i - 1] - (18 - 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 (7, 9]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-b9371054cfe1d371
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-b9371054cfe1d371
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-b9371054cfe1d371

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-b9371054cfe1d371

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-c3d762863d6658d7 basis_map (A038291) → 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