A099040 → x ↦ x−12

Proved for all n: all roots real, in (10, 12] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-24 + 2\,x\right)\,P_{n-1}(x) + \left(-24 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -24 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-24	2														-22
2	552	-94	4													462
3	-12672	3264	-280	8												-9680
4	290880	-100320	12964	-744	16											202796
5	-6676992	2885760	-498528	43032	-1856	32										-4248552
6	153266688	-79622784	17224416	-1986040	128736	-4448	64									89006632
7	-3518152704	2134867968	-554895360	80083968	-6931136	359744	-10368	128								-1864677760
8	80757264384	-56055656448	17014593024	-2949692160	319453456	-22131904	957888	-23680	256							39064764816
9	-1853738680320	1448577146880	-502874320896	101789991936	-13239752064	1147576480	-66284800	2460288	-53248	512						-818401915232
10	42551553982464	-36966478602240	14445676449792	-3343886650368	507767766144	-52851267040	3818735040	-189132416	6145024	-118272	1024					17145417309152
11	-976747567251456	933825265532928	-405663054004224	105695924060160	-18352865654784	2229944601600	-193466186880	11985031552	-519546880	15009792	-260096	2048				-359194338666240
12	22420704318455808	-23393002912874496	11182934635167744	-3238883652943872	632986424146944	-87940435806720	8905725513280	-662396482944	35913442816	-1384200192	36000768	-567296	4096			7525076269855936
13	-514654962028904448	581908177038606336	-303572873242607616	96751048655536128	-21017580861579264	3286318905907200	-380515206243840	33034393486208	-2150276405248	103648361472	-3596156928	85047296	-1228800	8192		-157649402486173312
14	11813622185050767360	-14388832694438461440	8134388874830020608	-2829071837277093888	676254596503560192	-117530272157276160	15315384477734400	-1520146889539456	115488705104896	-6683065795584	289972070400	-9147832320	198352896	-2646016	16384	3302735176758982272

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Technical details

Node id: n-6d02eb83391908c0 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-d87f139cafd00068

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:            -24,             2;
  2:            552,           -94,             4;
  3:         -12672,          3264,          -280,            8;
  4:         290880,       -100320,         12964,         -744,           16;
  5:       -6676992,       2885760,       -498528,        43032,        -1856,         32;
  6:      153266688,     -79622784,      17224416,     -1986040,       128736,      -4448,        64;
  7:    -3518152704,    2134867968,    -554895360,     80083968,     -6931136,     359744,    -10368,     128;
  8:    80757264384,  -56055656448,   17014593024,  -2949692160,    319453456,  -22131904,    957888,  -23680,    256;
  9: -1853738680320, 1448577146880, -502874320896, 101789991936, -13239752064, 1147576480, -66284800, 2460288, -53248, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(24-2*x)*P(n-1,x) - (24-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -24 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -24 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(24 - 2*x)*P[n - 1, x] - (24 - 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] = -24 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(24 - 2*x)*v[i - 1] - (24 - 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 (10, 12]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-1000187f132dd1ea
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-1000187f132dd1ea
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-1000187f132dd1ea

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-1000187f132dd1ea

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-ffeb0648278e3cee basis_map (A038327) → 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