A113214 → x ↦ x−9

Proved for all n: all roots real, in (5, 9] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-9 + x\right)\,P_{n-1}(x) + \left(-9 + x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -7 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-7	1														-6
2	54	-15	1													40
3	-423	173	-23	1												-272
4	3321	-1791	356	-31	1											1856
5	-26082	17460	-4615	603	-39	1										-12672
6	204849	-163782	54000	-9407	914	-47	1									86528
7	-1608903	1495665	-590787	128621	-16679	1289	-55	1								-590848
8	12636486	-13391001	6162966	-1609713	261099	-26943	1728	-63	1							4034560
9	-99248247	118085607	-62044947	18902007	-3680872	475306	-40711	2231	-71	1						-27549696
10	779505849	-1028863215	607632435	-211512627	48070251	-7455040	799210	-58495	2798	-79	1					188121088
11	-6122318418	8877256074	-5821065000	2279083068	-592115031	107206985	-13806225	1264875	-80807	3429	-87	1				-1284571136
12	48085313121	-75978348300	54769285944	-23821566534	6963973461	-1441812285	216815080	-23864435	1908461	-108159	4124	-95	1			8771600384
13	-377666952327	645872824737	-507635080722	242830572138	-78889209336	18383306130	-3161684995	406404895	-39048446	2770224	-141063	4883	-103	1		-59896233984
14	2966234752854	-5458631927139	4645686629439	-2423946845214	866336128479	-224398680480	43445323080	-6387734055	716800325	-61098570	3894516	-180031	5706	-111	1	408997068800

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Technical details

Node id: n-0e47b6d43ddfe6ee · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113214
Fingerprint: f6-5678377f8db7bf17

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:        -7,         1;
  2:        54,       -15,         1;
  3:      -423,       173,       -23,        1;
  4:      3321,     -1791,       356,      -31,        1;
  5:    -26082,     17460,     -4615,      603,      -39,      1;
  6:    204849,   -163782,     54000,    -9407,      914,    -47,      1;
  7:  -1608903,   1495665,   -590787,   128621,   -16679,   1289,    -55,    1;
  8:  12636486, -13391001,   6162966, -1609713,   261099, -26943,   1728,  -63,   1;
  9: -99248247, 118085607, -62044947, 18902007, -3680872, 475306, -40711, 2231, -71, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(9-x)*P(n-1,x) - (9-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -7 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -7 + x;
P[n_, x_] := P[n, x] = Expand[-(9 - x)*P[n - 1, x] - (9 - 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] = -7 + x; for(n = 2, N, my(i = n + 1); v[i] = -(9 - x)*v[i - 1] - (9 - 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 (5, 9]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-77c8319978ad9839
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-77c8319978ad9839
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-77c8319978ad9839

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-77c8319978ad9839

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-e67dd89b1bcbf6c7 basis_map (A038291) → 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