A113214 → x ↦ x−12

Proved for all n: all roots real, in (8, 12] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

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

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

\(P_{1}(x) = -10 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-10	1							-9
2	108	-21	1						88
3	-1176	338	-32	1					-869
4	12816	-4872	689	-43	1				8591
5	-139680	66048	-12418	1161	-54	1			-84942
6	1522368	-860976	201924	-25145	1754	-65	1		839861
7	-16592256	10921824	-3069000	477314	-44384	2468	-76	1	-8304109

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-d1b895351f517faa · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint: f6-bd8969e3a1059a26

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:         -10,          1;
  2:         108,        -21,          1;
  3:       -1176,        338,        -32,         1;
  4:       12816,      -4872,        689,       -43,         1;
  5:     -139680,      66048,     -12418,      1161,       -54,       1;
  6:     1522368,    -860976,     201924,    -25145,      1754,     -65,       1;
  7:   -16592256,   10921824,   -3069000,    477314,    -44384,    2468,     -76,    1;
  8:   180838656, -135800064,   44465760,  -8293104,    963729,  -71466,    3303,  -87,   1;
  9: -1970956800, 1662785280, -621639360, 135186240, -18847930, 1747321, -107722, 4259, -98, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(12-x)*P(n-1,x) - (12-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -10 + x.

%t Mathematica

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

by the cited steps: e-f8dba443f4d27d49
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-f8dba443f4d27d49
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-f8dba443f4d27d49

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-f8dba443f4d27d49

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1+2x,x(1+x)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
  - theorem_check ma_wang_interval e-5a8f629e61d609b9 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-253047f151171d84 basis_map (A038327) → Riordan array (1+2x,x(1+x)).

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