A099097 → x ↦ x−7

Proved for all n: all roots real, in (59/9, 7] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-21 + 3\,x\right)\,P_{n-1}(x) + \left(-7 + x\right)\,P_{n-2}(x)\)

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

\(P_{2}(x) = -21 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-21	3							-18
3	434	-125	9						318
4	-8967	3885	-561	27					-5616
5	185269	-107177	23248	-2241	81				99180
6	-3827880	2770362	-801927	116055	-8397	243			-1751544
7	79088597	-68725734	24881640	-4804001	521694	-30213	729		30932712
8	-1634065377	1657285791	-720307791	173914629	-25192743	2189457	-105705	2187	-546279552

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-fadb941bf197056d · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099097
Fingerprint: f6-f2ec9c9452b6f95c

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 -1)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:             1;
   2:           -21,            3;
   3:           434,         -125,             9;
   4:         -8967,         3885,          -561,           27;
   5:        185269,      -107177,         23248,        -2241,           81;
   6:      -3827880,      2770362,       -801927,       116055,        -8397,        243;
   7:      79088597,    -68725734,      24881640,     -4804001,       521694,     -30213,        729;
   8:   -1634065377,   1657285791,    -720307791,    173914629,    -25192743,    2189457,    -105705,     2187;
   9:   33761752738, -39145029007,   19855423770,  -5754620935,   1042335631, -120823641,    8752860,  -362313,     6561;
  10: -697558349859, 910095801447, -527699545863, 178475600751, -38802647226, 5623784412, -543351591, 33746139, -1222533, 19683;

%F Formula

The row polynomials satisfy P(n,x) = -(21-3*x)*P(n-1,x) - (7-x)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = -21 + 3*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := -21 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(21 - 3*x)*P[n - 1, x] - (7 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -21 + 3*x; for(n = 3, N, my(i = n); v[i] = -(21 - 3*x)*v[i - 1] - (7 - 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, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (59/9, 7]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-4e6408a181c26c62
Coefficients of one sign. Every nonzero row, n >= 1: 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-4e6408a181c26c62
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-4e6408a181c26c62

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-4e6408a181c26c62

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1, 3+x).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]
  - theorem_check ma_wang_interval e-09c939a271fe525a root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-ed12ae64460e2aac basis_map (A027466) → Riordan array (1, 3+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