A161198 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(\left(-1 + x + n\right)\,P_{n}(x) = \left(-3\,x + 2\,x^{2} - 3\,n + 4\,n x + 2\,n^{2}\right)\,P_{n-1}(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	2	2															4
2	4	10	4														18
3	18	54	40	8													120
4	120	398	380	136	16												1050
5	1050	3730	4144	1968	416	32											11340
6	11340	42426	52356	29680	8480	1184	64										145530
7	145530	567462	757880	486872	168800	32288	3200	128									2162160
8	2162160	8724894	12406012	8763160	3489584	819616	112448	8320	256								36486450
9	36486450	151590690	226933344	172867680	76527808	20850368	3544576	366080	20992	512							689188500
10	689188500	2936803050	4591521540	3721744352	1793105984	547635648	108829056	14028288	1130496	51712	1024						14404039650
11	14404039650	62764452630	101865085128	87010547368	44952316480	15046436800	3373810944	511536384	51755520	3345920	124928	2048					329983453800
12	329983453800	1466803672110	2459673293436	2197858398120	1204484559696	434917531840	107479737728	18485617920	2211230208	180364800	9563136	296960	4096				8222087723850
13	8222087723850	37210116622770	64230462079632	59698899806736	34421365969760	13252869689312	3550291672576	676104191488	92150699520	8924990976	599560192	26546176	696320	8192			221363900257500
14	221363900257500	1018303659005850	1803918340409220	1736102408878032	1046452176496480	425829416213344	122155191897024	25319421011456	3835956101120	424922015232	34019228672	1915277312	71884800	1613824	16384		6403741400306250
15	6403741400306250	29901927343695750	54226502634706200	53839637548731000	33752797708514976	14416292481138912	4387267236736384	977257360247168	161701588725248	19977154802176	1835221420032	123528996864	5913829376	190554112	3702784	32768	198089067316140000

Computed in your browser ?

Technical details

Node id: n-5aa82e9690458bf1 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-7fa978b055a9170d

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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): yes

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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:         2,          2;
   2:         4,         10,          4;
   3:        18,         54,         40,          8;
   4:       120,        398,        380,        136,         16;
   5:      1050,       3730,       4144,       1968,        416,        32;
   6:     11340,      42426,      52356,      29680,       8480,      1184,        64;
   7:    145530,     567462,     757880,     486872,     168800,     32288,      3200,      128;
   8:   2162160,    8724894,   12406012,    8763160,    3489584,    819616,    112448,     8320,     256;
   9:  36486450,  151590690,  226933344,  172867680,   76527808,  20850368,   3544576,   366080,   20992,   512;
  10: 689188500, 2936803050, 4591521540, 3721744352, 1793105984, 547635648, 108829056, 14028288, 1130496, 51712, 1024;

%F Formula

The row polynomials satisfy (n-1+x)*P(n,x) = (2*n^2-3*n+(4*n-3)*x+2*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 2 + 2*x.

%t Mathematica

P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 - 3*n + (4*n - 3)*x + 2*x^2)*P[n - 1, x])/(n - 1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2 + 2*x; for(n = 2, N, my(i = n); v[i] = ((2*n^2 - 3*n + (4*n - 3)*x + 2*x^2)*v[i - 1])/(n - 1 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, 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-3d3610943d28dca0

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-3d3610943d28dca0

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-6c4d7be2cbbf8954 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-84f434121cdb02d1 root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  
  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.