Real-rooted atlas

A126351 → Pₙ + Pₙ₊₁

nonnegative coefficients

Proved for all n: all roots real what this means

Recurrence
\(P_{n}(x) = \left(1 + n x\right)\,P_{n-2}(x) + \left(-x^{2}\right)\,P_{n-2}'(x) + \left(2\,x + n x\right)\,P_{n-1}(x) + \left(-x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2 + 2\,x\)
\(P_{1}(x) = 2 + 7\,x + 4\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0224
127413
221423847
3223747316188
423418035022732825
524737011951562697643937
6262679329072526734212312820270
7279114877982608241643283946433256111835

Show all 15 stored rows

Computed in your browser ?

Technical details
Node id
n-b2914cf224361a03 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-62f4a57d6f45f4d2

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 ?

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,   2;
  1: 2,   7,    4;
  2: 2,  14,   23,     8;
  3: 2,  23,   74,    73,     16;
  4: 2,  34,  180,   350,    227,     32;
  5: 2,  47,  370,  1195,   1562,    697,      64;
  6: 2,  62,  679,  3290,   7252,   6734,    2123,     128;
  7: 2,  79, 1148,  7798,  26082,  41643,   28394,    6433,    256;
  8: 2,  98, 1824, 16548,  78288, 192234,  230540,  117950,  19427,   512;
  9: 2, 119, 2760, 32250, 205254, 720069, 1348370, 1244315, 484922, 58537, 1024;
%F Formula
The row polynomials satisfy P(n,x) = (1+n*x)*P(n-2,x) - x^2*d/dx P(n-2,x) + (n+2)*x*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 2*x, P(1,x) = 2 + 7*x + 4*x^2.
%t Mathematica
P[0, x_] := 2 + 2*x;
P[1, x_] := 2 + 7*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 2, x] - x^2*D[P[n - 2, x], x] + (n + 2)*x*P[n - 1, x] - x^2*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 + 2*x; v[2] = 2 + 7*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + n*x)*v[i - 2] - x^2*deriv(v[i - 2], x) + (n + 2)*x*v[i - 1] - 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 - 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-68c0b23ee0fd1d24

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.