Real-rooted atlas

A008292 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(1 - 2\,x + n x\right)\,P_{n-2}(x) + \left(x - x^{2}\right)\,P_{n-2}'(x) + \left(n x\right)\,P_{n-1}(x) + \left(x - x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 2 + 5\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
1213
22518
321512130
423777271144
5283368328581840
6217714932718124812115760
72367548418035168104413248145360
827491890110385317180992527148555031403200

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-1567559463beb931 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-16501f2a611212d0

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 ?

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,    5,      1;
   3: 2,   15,     12,       1;
   4: 2,   37,     77,      27,        1;
   5: 2,   83,    368,     328,       58,        1;
   6: 2,  177,   1493,    2718,     1248,      121,        1;
   7: 2,  367,   5484,   18035,    16810,     4413,      248,       1;
   8: 2,  749,  18901,  103853,   171809,    92527,    14855,     503,      1;
   9: 2, 1515,  62448,  543426,  1466544,  1398588,   469800,   48342,   1014,    1;
  10: 2, 3049, 200477, 2658680, 11048468, 17034602, 10193306, 2251328, 153650, 2037, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n-2)*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + n*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + x, P(2,x) = 2 + 5*x + x^2.
%t Mathematica
P[1, x_] := 2 + x;
P[2, x_] := 2 + 5*x + x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (n - 2)*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + n*x*P[n - 1, x] + (x - x^2)*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; v[2] = 2 + 5*x + x^2; for(n = 3, N, my(i = n); v[i] = (1 + (n - 2)*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + 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, 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-8f0a401a806157c3

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

Same fingerprint (1)

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

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.