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	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

Show all 15 stored rows

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.