Real-rooted atlas

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 ?

nx0x1x2x3x4x5x6x7x8Pn(1)
1213
228111
325019172
42302259351599
52181431518195615843
62108863647516255199582165696
72653184121112965555815041271131836377
823919104598539515699115177051654527630149111838379

Show all 15 stored rows

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

No candidate equalities or related sequences.

References

  1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
  2. [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
  3. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.