A028338 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	2	1															3
1	4	5	1														10
2	18	27	10	1													56
3	120	199	95	17	1												432
4	1050	1865	1036	246	26	1											4224
5	11340	21213	13089	3710	530	37	1										49920
6	145530	283731	189470	60859	10550	1009	50	1									691200
7	2162160	4362447	3101503	1095395	218099	25613	1757	65	1								10967040
8	36486450	75795345	56733336	21608460	4782988	651574	55384	2860	82	1							196116480
9	689188500	1468401525	1147880385	465218044	112069124	17113614	1700454	109596	4416	101	1						3901685760
10	14404039650	31382226315	25466271282	10876318421	2809519780	470201150	52715796	3996378	202170	6535	122	1					85465497600
11	329983453800	733401836055	614918323359	274732299765	75280284981	13591172870	1679370902	144418890	8637618	352275	9339	145	1				2043740160000
12	8222087723850	18605058311385	16057615519908	7462362475842	2151335373110	414152177791	55473307384	5282063996	359963670	17431623	585508	12962	170	1			52973744947200
13	221363900257500	509151829502925	450979585102305	217012801109754	65403261031030	13307169256667	1908674873391	197807976652	14984203520	829925811	33221903	935194	17550	197	1		1479340877414400
14	6403741400306250	14950963671847875	13556625658676550	6729954693591375	2109549856782186	450509140035591	68551050574006	7634823126931	631646830958	39017880473	1792208418	60316893	1443806	23261	226	1	44278202813644800

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Technical details

Node id: n-8cae8db3d01b43b1 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-c0a519a80d51f885

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 1)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         2,          1;
  1:         4,          5,          1;
  2:        18,         27,         10,         1;
  3:       120,        199,         95,        17,         1;
  4:      1050,       1865,       1036,       246,        26,        1;
  5:     11340,      21213,      13089,      3710,       530,       37,       1;
  6:    145530,     283731,     189470,     60859,     10550,     1009,      50,      1;
  7:   2162160,    4362447,    3101503,   1095395,    218099,    25613,    1757,     65,    1;
  8:  36486450,   75795345,   56733336,  21608460,   4782988,   651574,   55384,   2860,   82,   1;
  9: 689188500, 1468401525, 1147880385, 465218044, 112069124, 17113614, 1700454, 109596, 4416, 101, 1;

%F Formula

The row polynomials satisfy (2*n+x)*P(n,x) = (4*n^2+2*n-2+(4*n+1)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.

%t Mathematica

P[0, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n - 2 + (4*n + 1)*x + x^2)*P[n - 1, x])/(2*n + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 2*n - 2 + (4*n + 1)*x + x^2)*v[i - 1])/(2*n + 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-3733e878156287d8

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-3733e878156287d8

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-59b849691f316c6f root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - parent_provers root_domain e-ef36674c01230a86 root: theorem check
    
    Parent-row provers of polytool polytool ?
- premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
  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.