A113214 → x ↦ x−6

Proved for all n: all roots real, in (2, 6] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-6 + x\right)\,P_{n-1}(x) + \left(-6 + x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -4 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-4	1							-3
2	18	-9	1						10
3	-84	62	-14	1					-35
4	396	-384	131	-19	1				125
5	-1872	2244	-1024	225	-24	1			-450
6	8856	-12636	7218	-2129	344	-29	1		1625
7	-41904	69336	-47556	17618	-3824	488	-34	1	-5875

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-22896c6867f82de3 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint: f6-e4f4f1515458a372

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:      -4,       1;
  2:      18,      -9,        1;
  3:     -84,      62,      -14,       1;
  4:     396,    -384,      131,     -19,       1;
  5:   -1872,    2244,    -1024,     225,     -24,     1;
  6:    8856,  -12636,     7218,   -2129,     344,   -29,     1;
  7:  -41904,   69336,   -47556,   17618,   -3824,   488,   -34,   1;
  8:  198288, -373248,   298728, -133272,   36369, -6234,   657, -39,   1;
  9: -938304, 1979856, -1810944,  945096, -310924, 67021, -9484, 851, -44, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(6-x)*P(n-1,x) - (6-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -4 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -4 + x;
P[n_, x_] := P[n, x] = Expand[-(6 - x)*P[n - 1, x] - (6 - 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] = -4 + x; for(n = 2, N, my(i = n + 1); v[i] = -(6 - x)*v[i - 1] - (6 - 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 (2, 6]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-defa6799aaae8d2a
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-defa6799aaae8d2a
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-defa6799aaae8d2a

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-defa6799aaae8d2a

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-2011332c4965ce14 basis_map (A038255) → 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