A166973 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(1 + x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
1	2	1															3
2	2	8	1														11
3	2	50	19	1													72
4	2	302	259	35	1												599
5	2	1814	3151	819	56	1											5843
6	2	10886	36475	16255	1995	82	1										65696
7	2	65318	412111	296555	58150	4127	113	1									836377
8	2	391910	4598539	5156991	1517705	165452	7630	149	1								11838379
9	2	2351462	50975839	87110395	37028796	5819457	401982	12994	190	1							183701118
10	2	14108774	563085691	1444742159	864715111	188334678	18280899	869766	20784	236	1						3094158101
11	2	84652646	6208051375	23678960235	19603759490	5761416739	755042547	49592475	1721910	31640	287	1					56143229347
12	2	507915878	68373217771	385071415135	435357909525	169400594704	29167735696	2540371647	120190785	3177350	46277	343	1				1090542575114
13	2	3047495270	752613311359	6229515859931	9527587515160	4839773371829	1073600401280	120621114988	7468193832	266348885	5537477	65485	404	1			22554499215903
14	2	18284971622	8281793920219	100424867070255	206308853678291	135361695182714	38121385811509	5415960540848	426817062100	19720242542	548760212	9204637	90129	470	1		494379936535551
15	2	109709829734	91118018094031	1615079667044299	4432910794314366	3725712928428855	1317124655339493	233095965282037	22915460086948	1333948219032	47707013354	1064219884	14702506	121149	541	1	11439449932696232

Computed in your browser ?

Technical details

Node id: n-7b9e0785bd039123 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-58d17f888e14c685

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): 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,        1;
   2: 2,        8,         1;
   3: 2,       50,        19,          1;
   4: 2,      302,       259,         35,         1;
   5: 2,     1814,      3151,        819,        56,         1;
   6: 2,    10886,     36475,      16255,      1995,        82,        1;
   7: 2,    65318,    412111,     296555,     58150,      4127,      113,      1;
   8: 2,   391910,   4598539,    5156991,   1517705,    165452,     7630,    149,     1;
   9: 2,  2351462,  50975839,   87110395,  37028796,   5819457,   401982,  12994,   190,   1;
  10: 2, 14108774, 563085691, 1444742159, 864715111, 188334678, 18280899, 869766, 20784, 236, 1;

%F Formula

The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2 + x.

%t Mathematica

P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 5*x*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 + x; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + 5*x*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-61c0915f6ff6bb4a

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-61c0915f6ff6bb4a

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m…
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-cdec5b72bd920a86 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m…
  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.