A102365 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

\(P_{0}(x) = 2\)

\(P_{1}(x) = 2 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	2	1														3
2	2	6	1													9
3	2	23	16	1												42
4	2	76	144	38	1											261
5	2	237	1006	683	84	1										2013
6	2	722	6157	8676	2768	178	1									18504
7	2	2179	34938	90640	58970	10179	368	1								197277
8	2	6552	189400	838828	975900	345164	35144	750	1							2391741
9	2	19673	997780	7167938	13733458	8620019	1827526	116315	1516	1						32484228
10	2	59038	5161041	57998064	173020470	176301342	66522771	9020052	373968	3050	1					488459799
11	2	177135	26380828	451597779	2014757022	3137644209	1914319218	466121886	42322098	1178339	6120	1				8054504637
12	2	531428	133808344	3420669730	22145606865	50471935848	46681642812	18413013576	3041180610	191304380	3660488	12262	1			144503366346
13	2	1594309	675286002	25394522069	233117147650	752647205718	1007667215616	601117114128	161720934426	18798736625	840912478	11257371	24548	1		2801991950943
14	2	4782954	3396757453	185753335988	2374373413314	10589999682928	19825996346346	17033262383232	6939119887962	1324502371466	111462716073	3618915076	34373552	49122	1	58391525015469

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Technical details

Node id: n-66d9da858bbfa63a · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-2f693965441175ca

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): yes
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: 2;
  1: 2,     1;
  2: 2,     6,      1;
  3: 2,    23,     16,       1;
  4: 2,    76,    144,      38,        1;
  5: 2,   237,   1006,     683,       84,       1;
  6: 2,   722,   6157,    8676,     2768,     178,       1;
  7: 2,  2179,  34938,   90640,    58970,   10179,     368,      1;
  8: 2,  6552, 189400,  838828,   975900,  345164,   35144,    750,    1;
  9: 2, 19673, 997780, 7167938, 13733458, 8620019, 1827526, 116315, 1516, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+(n-2)*x)*P(n-2,x) + (2*x-x^2)*d/dx P(n-2,x) + n*x*P(n-1,x) + (2*x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 2)*x)*P[n - 2, x] + (2*x - x^2)*D[P[n - 2, x], x] + n*x*P[n - 1, x] + (2*x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + (n - 2)*x)*v[i - 2] + (2*x - x^2)*deriv(v[i - 2], x) + n*x*v[i - 1] + (2*x - x^2)*deriv(v[i - 1], x)); 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 (-inf, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-9fcf9816b8c5606e

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

obreschkoff_combination e-9fcf9816b8c5606e

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, …
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-1ce099339e80292c root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, …
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  
  no recorded property covers this requirement

Related sequences ?

No candidate equalities or related sequences.

References

[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.