A282629 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence

\(P_{n}(x) = \left(1 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	2	3								5
1	2	18	9							29
2	2	78	117	27						224
3	2	318	1053	621	81					2075
4	2	1278	8325	9369	2916	243				22133
5	2	5118	62109	118665	66015	12636	729			265274
6	2	20478	450117	1372977	1214190	400221	51759	2187		3511931
7	2	81918	3212253	15080121	19903401	10046106	2184084	203391	6561	50717837

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-694eb1ea263db23a · revision 1 · JSON
Definition source: derived:obreschkoff_combination
Fingerprint: f6-bf272cba03b71d47

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: 2,       3;
  1: 2,      18,         9;
  2: 2,      78,       117,         27;
  3: 2,     318,      1053,        621,         81;
  4: 2,    1278,      8325,       9369,       2916,        243;
  5: 2,    5118,     62109,     118665,      66015,      12636,        729;
  6: 2,   20478,    450117,    1372977,    1214190,     400221,      51759,      2187;
  7: 2,   81918,   3212253,   15080121,   19903401,   10046106,    2184084,    203391,     6561;
  8: 2,  327678,  22731525,  160437969,  303984576,  220447899,   71635914,  11026854,   774198,   19683;
  9: 2, 1310718, 160103709, 1672574265, 4433113395, 4439120112, 2022426063, 457498530, 52435512, 2873718, 59049;

%F Formula

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

%t Mathematica

P[0, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 1, x] + 3*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] = 2 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + 3*x)*v[i - 1] + 3*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-b6db328b817fb97a

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-b6db328b817fb97a

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?
- premise (sequence): Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check liu_wang e-949f12f88dca835e root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
- premise (sequence): Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].
  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.