A099040 → x ↦ x−7

Proved for all n: all roots real, in (5, 7] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-14 + 2\,x\right)\,P_{n-1}(x) + \left(-14 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -14 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-14	2														-12
2	182	-54	4													132
3	-2352	1064	-160	8												-1440
4	30380	-18480	4204	-424	16											15696
5	-392392	299880	-91448	13912	-1056	32										-171072
6	5068168	-4663624	1784216	-363320	41536	-2528	64									1864512
7	-65460864	70443968	-32426240	8277248	-1265536	115904	-5888	128								-20321280
8	845497744	-1041710208	560549024	-172079040	32963856	-4035264	308288	-13440	256							221481216
9	-10920516320	15157801120	-9336251456	3349470656	-771380064	118267680	-12072320	791168	-30208	512						-2413919232
10	141050260064	-217775309920	151092015872	-62034887488	16692610144	-3076086240	393161280	-34416256	1974784	-67072	1024					26309256192
11	-1821816412416	3096904610688	-2389815719424	1105107364480	-340268054784	73251920000	-11250882560	1232929152	-94474240	4820992	-147456	2048				-286744043520
12	23530726132928	-43669342515456	37100390451264	-19080462084992	6616201178944	-1629612561920	292359765440	-38494623104	3692018176	-251553792	11557888	-321536	4096			3125217447936
13	-303924736087168	611431950107776	-567092922055296	321076115550848	-123813773179264	34340915235200	-7048245644160	1083881481088	-124889003008	10649347072	-653211648	27293696	-696320	8192		-34061680852992
14	3925516139359360	-8509464526200960	8555420657641088	-5287924211730048	2244757314936192	-692353381426560	160005007648640	-28147187769216	3787531503616	-387963075584	29778739200	-1660917760	63635456	-1499136	16384	371237560860672

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Technical details

Node id: n-fd1f27ef1d8123fd · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-cacf9f85ae71fd07

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:          -14,           2;
  2:          182,         -54,           4;
  3:        -2352,        1064,        -160,          8;
  4:        30380,      -18480,        4204,       -424,         16;
  5:      -392392,      299880,      -91448,      13912,      -1056,        32;
  6:      5068168,    -4663624,     1784216,    -363320,      41536,     -2528,        64;
  7:    -65460864,    70443968,   -32426240,    8277248,   -1265536,    115904,     -5888,    128;
  8:    845497744, -1041710208,   560549024, -172079040,   32963856,  -4035264,    308288, -13440,    256;
  9: -10920516320, 15157801120, -9336251456, 3349470656, -771380064, 118267680, -12072320, 791168, -30208, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(14-2*x)*P(n-1,x) - (14-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -14 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -14 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(14 - 2*x)*P[n - 1, x] - (14 - 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] = -14 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(14 - 2*x)*v[i - 1] - (14 - 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 (5, 7]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-5e00209341210844
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-5e00209341210844
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-5e00209341210844

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-5e00209341210844

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-3f03a2606d2e6662 basis_map (A027466) → 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