A163936 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

\(P_{1}(x) = 3 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	2															2
1	3	1														4
2	8	9	1													18
3	30	66	23	1												120
4	144	502	350	53	1											1050
5	840	4152	4728	1504	115	1										11340
6	5760	37692	62540	33572	5724	241	1									145530
7	45360	375120	843444	676140	201410	20190	495	1								2162160
8	403200	4075056	11811600	13084084	5961300	1082450	67754	1005	1							36486450
9	3991680	48072960	173213208	251344968	161122884	45827500	5393420	219852	2027	1						689188500
10	43545600	612695520	2669668128	4881068856	4158052472	1693149304	319695880	25462752	697064	4073	1					14404039650
11	518918400	8396075520	43295783040	96841954176	105183128112	57689783088	15863330504	2076726680	115571414	2174698	8167	1				329983453800
12	6706022400	123163804800	738717738624	1975279452048	2651045906448	1874230041088	703269007168	136374903000	12784694152	509429902	6707862	16357	1			8222087723850
13	93405312000	1926328642560	13248833454720	41572657717632	67278240613296	59312562106896	28987630112496	7769155569616	1097346330192	75523311504	2196529504	20524344	32739	1		221363900257500
14	1394852659200	32007369542400	249459175560960	904712831275392	1731152861411328	1854665594155152	1140047382635792	402000419964912	79476061343728	8383460380196	432015836148	9313034908	62440628	65505	1	6403741400306250

Computed in your browser ?

Technical details

Node id: n-770bc0f5eb42b52a · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-78696c1f329edd89

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 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): 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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:       2;
  1:       3,        1;
  2:       8,        9,         1;
  3:      30,       66,        23,         1;
  4:     144,      502,       350,        53,         1;
  5:     840,     4152,      4728,      1504,       115,        1;
  6:    5760,    37692,     62540,     33572,      5724,      241,       1;
  7:   45360,   375120,    843444,    676140,    201410,    20190,     495,      1;
  8:  403200,  4075056,  11811600,  13084084,   5961300,  1082450,   67754,   1005,    1;
  9: 3991680, 48072960, 173213208, 251344968, 161122884, 45827500, 5393420, 219852, 2027, 1;

%F Formula

The row polynomials satisfy P(n,x) = (n-1+(n-2)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+n*x)*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(n - 1 + (n - 2)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n + n*x)*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (n - 1 + (n - 2)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + n*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 - 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-e0e39e6ac529ed26

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-e0e39e6ac529ed26

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-9a9d643a8c105142 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle related to the o.g.f.s. of the right-hand columns of A130534 (E(x,m=1,n)).
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
  
  no recorded property covers this requirement

Related sequences ?

Same fingerprint (1)

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

A112007 → 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.