A201637 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(1 - 6\,x + 2\,n x\right)\,P_{n-2}(x) + \left(x - x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x + 2\,n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	2															2
2	2	2														4
3	2	10	6													18
4	2	30	64	24												120
5	2	74	386	468	120											1050
6	2	166	1780	4844	3828	720										11340
7	2	354	7062	36520	61848	34704	5040									145530
8	2	734	25560	227920	702160	819288	346176	40320								2162160
9	2	1498	87210	1258300	6409520	13225368	11367432	3774240	362880							36486450
10	2	3030	286108	6388660	50530500	167797448	249931200	165920832	44701920	3628800						689188500
11	2	6098	913886	30570064	359134720	1803741688	4241599992	4804350864	2551820256	571985280	39916800					14404039650
12	2	12238	2865664	140120280	2365852496	17197345088	60044093872	105822596976	94707271440	41356658304	7867635840	479001600				329983453800
13	2	24522	8870322	622135652	14721300552	149872381008	743761445168	1919369075328	2642065263648	1923867963648	705757155840	115815087360	6227020800			8222087723850
14	2	49094	27207684	2697089084	87685870004	1218999932640	8322552195776	30118098708416	60044238938016	66611454801696	40387507492608	12666240103680	1817219577600	87178291200		221363900257500
15	2	98242	82915878	11482356728	504842058568	9393120840384	86026216980704	422665497881632	1169241846034272	1861899595830432	1706114064327264	877574157237504	238719264099840	30283555276800	1307674368000	6403741400306250

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Technical details

Node id: n-55e6206734b15203 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-398a8acb4f052aaf

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 -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): 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 2;
   2: 2,    2;
   3: 2,   10,      6;
   4: 2,   30,     64,      24;
   5: 2,   74,    386,     468,      120;
   6: 2,  166,   1780,    4844,     3828,       720;
   7: 2,  354,   7062,   36520,    61848,     34704,      5040;
   8: 2,  734,  25560,  227920,   702160,    819288,    346176,     40320;
   9: 2, 1498,  87210, 1258300,  6409520,  13225368,  11367432,   3774240,   362880;
  10: 2, 3030, 286108, 6388660, 50530500, 167797448, 249931200, 165920832, 44701920, 3628800;

%F Formula

The row polynomials satisfy P(n,x) = (1+(2*n-6)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (2*n-2)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2, P(2,x) = 2 + 2*x.

%t Mathematica

P[1, x_] := 2;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n - 6)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (2*n - 2)*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 2; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = (1 + (2*n - 6)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (2*n - 2)*x*v[i - 1] + (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, 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-ecc7f59b5400851a

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-ecc7f59b5400851a

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-d7adf0401ce51e02 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle of second-order Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
  
  no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.

A008517 → Pₙ + Pₙ₊₁

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.