A356546 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

Recurrence

\(\left(-2 - 2\,x + 3\,n + 2\,n x + 5\,n^{2} + 4\,n^{2} x\right)\,P_{n}(x) = \left(-6 - 10\,x - 4\,x^{2} + 2\,n + 2\,n x + 20\,n^{2} + 36\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	3	2															5
1	8	14	6														28
2	26	72	66	20													184
3	90	340	480	300	70												1280
4	322	1540	2940	2800	1330	252											9184
5	1176	6804	16380	21000	15120	5796	924										67200
6	4356	29568	85932	138600	133980	77616	24948	3432									498432
7	16302	126984	432432	840840	1021020	792792	384384	106392	12870								3734016
8	61490	540540	2110680	4804800	7027020	6846840	4444440	1853280	450450	48620							28188160
9	233376	2285140	10064340	26254800	44924880	52684632	42882840	23921040	8751600	1896180	184756						214083584
10	890188	9607312	47112780	138567000	271591320	372468096	364708344	254963280	124710300	40646320	7944508	705432					1633914880
11	3409588	40209624	217273056	711310600	1571349780	2467601136	2824549728	2374484112	1454953500	633713080	186234048	33155304	2704156				12520947712
12	13104756	167657672	989721096	3569485920	8774986220	15527263752	20346069744	19989121152	14724129420	8031343320	3153045896	843696672	137911956	10400600			96277938176
13	50517200	696840200	4461857400	17577014000	47593145600	93699005400	138317579400	155527600800	133855722000	87749862200	43131288200	15413689200	3785818400	572033000	40116600		742472089600
14	195234120	2888395200	19937950200	85180914000	251892131400	546131345760	896833337400	1135861412400	1118651391000	856676620800	505974629160	226337857200	74229082200	16848972000	2366879400	155117520	5740161269760

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Technical details

Node id: n-fb0ded5b295bc8c5 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-2d051ac784554c2b

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:      3,       2;
  1:      8,      14,        6;
  2:     26,      72,       66,       20;
  3:     90,     340,      480,      300,       70;
  4:    322,    1540,     2940,     2800,     1330,      252;
  5:   1176,    6804,    16380,    21000,    15120,     5796,      924;
  6:   4356,   29568,    85932,   138600,   133980,    77616,    24948,     3432;
  7:  16302,  126984,   432432,   840840,  1021020,   792792,   384384,   106392,   12870;
  8:  61490,  540540,  2110680,  4804800,  7027020,  6846840,  4444440,  1853280,  450450,   48620;
  9: 233376, 2285140, 10064340, 26254800, 44924880, 52684632, 42882840, 23921040, 8751600, 1896180, 184756;

%F Formula

The row polynomials satisfy (5*n^2+3*n-2+(4*n^2+2*n-2)*x)*P(n,x) = (20*n^2+2*n-6+(36*n^2+2*n-10)*x+(16*n^2-4)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((20*n^2 + 2*n - 6 + (36*n^2 + 2*n - 10)*x + (16*n^2 - 4)*x^2)*P[n - 1, x])/(5*n^2 + 3*n - 2 + (4*n^2 + 2*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] = 3 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((20*n^2 + 2*n - 6 + (36*n^2 + 2*n - 10)*x + (16*n^2 - 4)*x^2)*v[i - 1])/(5*n^2 + 3*n - 2 + (4*n^2 + 2*n - 2)*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-9dcb6fc37a08fff7

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-9dcb6fc37a08fff7

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-883c9c594ff729de root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
  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.