A113214 → x ↦ x−10

Proved for all n: all roots real, in (6, 10] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-10 + x\right)\,P_{n-1}(x) + \left(-10 + x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -8 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-8	1							-7
2	70	-17	1						54
3	-620	222	-26	1					-423
4	5500	-2600	455	-35	1				3321
5	-48800	28660	-6668	769	-44	1			-26082
6	433000	-303900	88190	-13553	1164	-53	1		204849
7	-3842000	3136600	-1090460	209362	-23984	1640	-62	1	-1608903

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-0e6280284de0b3d7 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint: f6-f13aebf826d8afd9

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:         -8,         1;
  2:         70,       -17,          1;
  3:       -620,       222,        -26,        1;
  4:       5500,     -2600,        455,      -35,        1;
  5:     -48800,     28660,      -6668,      769,      -44,      1;
  6:     433000,   -303900,      88190,   -13553,     1164,    -53,      1;
  7:   -3842000,   3136600,   -1090460,   209362,   -23984,   1640,    -62,    1;
  8:   34090000, -31736000,   12855400, -2960360,   424009, -38690,   2197,  -71,   1;
  9: -302480000, 316242000, -146248800, 39274920, -6751248, 770525, -58400, 2835, -80, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(10-x)*P(n-1,x) - (10-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -8 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -8 + x;
P[n_, x_] := P[n, x] = Expand[-(10 - x)*P[n - 1, x] - (10 - 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] = -8 + x; for(n = 2, N, my(i = n + 1); v[i] = -(10 - x)*v[i - 1] - (10 - 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 (6, 10]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-e470a688ac88b1cf
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-e470a688ac88b1cf
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-e470a688ac88b1cf

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-e470a688ac88b1cf

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-0902c3fe08444c8c basis_map (A038303) → 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