A030528 → x ↦ x−7

Proved for all n: all roots real, in (3, 7] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

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

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

\(P_{1}(x) = -6 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-6	1														-5
2	35	-12	1													24
3	-203	106	-18	1												-114
4	1176	-826	213	-24	1											540
5	-6811	6013	-2085	356	-30	1										-2556
6	39445	-41944	18291	-4196	535	-36	1									12096
7	-228438	284151	-149373	43086	-7375	750	-42	1								-57240
8	1322951	-1884442	1159781	-403312	86770	-11838	1001	-48	1							270864
9	-7661591	12296550	-8673147	3531990	-915991	157011	-17801	1288	-54	1						-1281744
10	44370480	-79223396	63005670	-29414112	8933225	-1845432	262773	-25480	1611	-60	1					6065280
11	-256962223	505196811	-447254507	235507377	-82002760	19836181	-3403225	414316	-35091	1970	-66	1				-28701216
12	1488142201	-3194405648	3115715274	-1826901692	717580010	-199004778	39973913	-5862304	623196	-46850	2365	-72	1			135815616
13	-8618259846	20055641837	-21368434206	13808220972	-6040435065	1889757429	-435163413	74706604	-9564723	902265	-60973	2796	-78	1		-642686400
14	49910823515	-125158770968	144630268713	-102121953892	49241304665	-17158123612	4457079151	-877099600	131434989	-14929432	1265671	-77676	3263	-84	1	3041224704

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Technical details

Node id: n-cc7e0c23bafd90c3 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A030528
Fingerprint: f6-4c9fef36566d861c

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:       -6,        1;
  2:       35,      -12,        1;
  3:     -203,      106,      -18,       1;
  4:     1176,     -826,      213,     -24,       1;
  5:    -6811,     6013,    -2085,     356,     -30,      1;
  6:    39445,   -41944,    18291,   -4196,     535,    -36,      1;
  7:  -228438,   284151,  -149373,   43086,   -7375,    750,    -42,    1;
  8:  1322951, -1884442,  1159781, -403312,   86770, -11838,   1001,  -48,   1;
  9: -7661591, 12296550, -8673147, 3531990, -915991, 157011, -17801, 1288, -54, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(7-x)*P(n-1,x) - (7-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -6 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -6 + x;
P[n_, x_] := P[n, x] = Expand[-(7 - x)*P[n - 1, x] - (7 - 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] = -6 + x; for(n = 2, N, my(i = n + 1); v[i] = -(7 - 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 - 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 (3, 7]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-742e5241e9a4e627
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-742e5241e9a4e627
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-742e5241e9a4e627

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-742e5241e9a4e627

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = binomial(k,n-k).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0]
  - theorem_check ma_wang_interval e-f811a014585c6352 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-1781530cbfe1e9bf basis_map (A027466) → Triangle read by rows: T(n,k) = binomial(k,n-k).

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