A075497 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	3	1								4
1	6	7	1							14
2	12	34	13	1						60
3	24	148	112	21	1					306
4	48	616	820	280	31	1				1796
5	96	2512	5536	3060	590	43	1			11838
6	192	10144	35728	30016	8960	1106	57	1		86204
7	384	40768	224512	275856	119616	22232	1904	73	1	685346

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-b4652f0ae8d5ecf9 · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-07545fccb5ddeafe

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): 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:    3,      1;
  1:    6,      7,       1;
  2:   12,     34,      13,        1;
  3:   24,    148,     112,       21,        1;
  4:   48,    616,     820,      280,       31,       1;
  5:   96,   2512,    5536,     3060,      590,      43,       1;
  6:  192,  10144,   35728,    30016,     8960,    1106,      57,     1;
  7:  384,  40768,  224512,   275856,   119616,   22232,    1904,    73,    1;
  8:  768, 163456, 1387840,  2431360,  1472016,  386400,   48888,  3072,   91,   1;
  9: 1536, 654592, 8490496, 20838720, 17151520, 6108816, 1070832, 98040, 4710, 111, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + 2*x*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] = 3 + x; for(n = 1, N, my(i = n + 1); v[i] = (2 + x)*v[i - 1] + 2*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 - 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-b1f98f91a12826e9

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-b1f98f91a12826e9

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Stirling2 triangle with scaled diagonals (powers of 2).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-ca5f07049cdde3bf root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Stirling2 triangle with scaled diagonals (powers of 2).
  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.