A113953 → x ↦ x−12

Proved for all n: all roots real, in (4, 12] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-12 + x\right)\,P_{n-1}(x) + \left(-24 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -12 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-12	1														-11
2	120	-22	1													99
3	-1152	336	-32	1												-847
4	10944	-4416	652	-42	1											7139
5	-103680	53568	-10800	1068	-52	1										-59895
6	981504	-618624	158688	-21304	1584	-62	1									501787
7	-9289728	6912000	-2156544	367104	-36928	2200	-72	1								-4201967
8	87920640	-75423744	27744768	-5733120	729616	-58672	2916	-82	1							35182323
9	-832094208	808538112	-342779904	83418624	-12868032	1307024	-87536	3732	-92	1						-294562279
10	7875035136	-8548540416	4105175040	-1150718976	208857984	-25684960	2170128	-124520	4648	-102	1					2466173963
11	-74530160640	89388417024	-47966846976	15226195968	-3181344768	459972864	-47011584	3399728	-170624	5664	-112	1				-20647543455
12	705361084416	-926276124672	549369298944	-194853593088	46088303616	-7666864128	920658880	-80479584	5086624	-226848	6780	-122	1			172867150819
13	-6675609157632	9526292250624	-6188726550528	2426250018816	-641108570112	120688634880	-16666546944	1710797248	-130624640	7331616	-294192	7996	-132	1		-1447292702999
14	63178643865600	-97349767004160	68753595432960	-29528501944320	8623726387200	-1813190842368	283257656832	-33423286144	2995254784	-202986432	10245504	-373656	9312	-142	1	12117142414971

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Technical details

Node id: n-91b6a5a9e4e04cbc · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113953
Fingerprint: f6-0d3e004da859c95a

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:        -12,         1;
  2:        120,       -22,          1;
  3:      -1152,       336,        -32,        1;
  4:      10944,     -4416,        652,      -42,         1;
  5:    -103680,     53568,     -10800,     1068,       -52,       1;
  6:     981504,   -618624,     158688,   -21304,      1584,     -62,      1;
  7:   -9289728,   6912000,   -2156544,   367104,    -36928,    2200,    -72,    1;
  8:   87920640, -75423744,   27744768, -5733120,    729616,  -58672,   2916,  -82,   1;
  9: -832094208, 808538112, -342779904, 83418624, -12868032, 1307024, -87536, 3732, -92, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(12-x)*P(n-1,x) - (24-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -12 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -12 + x;
P[n_, x_] := P[n, x] = Expand[-(12 - 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] = -12 + x; for(n = 2, N, my(i = n + 1); v[i] = -(12 - 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 (4, 12]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-0ef3f7736abad08b
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-0ef3f7736abad08b
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-0ef3f7736abad08b

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-0ef3f7736abad08b

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-39068ae2f1d878f6 basis_map (A038327) → 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