A113953 → x ↦ x−15

Proved for all n: all roots real, in (7, 15] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-15 + x\right)\,P_{n-1}(x) + \left(-30 + 2\,x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = -15 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-15	1														-14
2	195	-28	1													168
3	-2475	555	-41	1												-1960
4	31275	-9570	1084	-54	1											22736
5	-394875	153225	-23490	1782	-67	1										-263424
6	4984875	-2343600	453915	-46432	2649	-80	1									3051328
7	-62926875	34752375	-8141175	1049955	-80593	3685	-93	1								-35342720
8	794356875	-503934750	138565350	-21589710	2086516	-128170	4890	-106	1							409360896
9	-10027546875	7184953125	-2268675000	414630000	-48369750	3737330	-191360	6264	-119	1						-4741456384
10	126582496875	-101095087500	36050248125	-7563303000	1034401350	-96411568	6204690	-272360	7807	-132	1					54918284288
11	-1597911046875	1407405121875	-559418653125	132523543125	-20798970750	2271715470	-176266458	9719450	-373367	9519	-145	1				-636095201280
12	20171190796875	-19422970256250	8514987300000	-2248272213750	398349457875	-49913553060	4536748504	-301478028	14541025	-496578	11400	-158	1			7367620857856
13	-254630530546875	266097768890625	-127550409918750	37144526906250	-7334497873125	1037303348175	-108133355940	8414802508	-488953493	20957391	-644190	13450	-171	1		-85336026374144
14	3214322234296875	-3623065574625000	1885058358159375	-600240172500000	130714966839375	-20599942588200	2423374126035	-216237555712	14709918097	-759334968	29285085	-818400	15669	-184	1	988410985218048

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Technical details

Node id: n-eb460955a5ddb26b · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A113953
Fingerprint: f6-62a4b2b77a02531a

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:          -15,          1;
  2:          195,        -28,           1;
  3:        -2475,        555,         -41,         1;
  4:        31275,      -9570,        1084,       -54,         1;
  5:      -394875,     153225,      -23490,      1782,       -67,       1;
  6:      4984875,   -2343600,      453915,    -46432,      2649,     -80,       1;
  7:    -62926875,   34752375,    -8141175,   1049955,    -80593,    3685,     -93,    1;
  8:    794356875, -503934750,   138565350, -21589710,   2086516, -128170,    4890, -106,    1;
  9: -10027546875, 7184953125, -2268675000, 414630000, -48369750, 3737330, -191360, 6264, -119, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(15-x)*P(n-1,x) - (30-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -15 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -15 + x;
P[n_, x_] := P[n, x] = Expand[-(15 - x)*P[n - 1, x] - (30 - 2*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] = -15 + x; for(n = 2, N, my(i = n + 1); v[i] = -(15 - x)*v[i - 1] - (30 - 2*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 (7, 15]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-0ffa0091fcab81ab
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-0ffa0091fcab81ab
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-0ffa0091fcab81ab

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-0ffa0091fcab81ab

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): A Jacobsthal triangle.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-8, 0]
  - theorem_check ma_wang_interval e-2d5b16bf2955a407 root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-0db30c1d149f8e9a basis_map (A027467) → A Jacobsthal triangle.

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