Real-rooted atlas

A028326 → Pₙ + Pₙ₊₁

nonnegative coefficients only rational roots

Proved for all n: all roots real what this means

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

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0426
146212
24108224
34141810248
4418322812296
5422506040142192
64267211010054162384
74309818221015470182768

Show all 15 stored rows

Generated b-files: A091629 (row sums) ?

Computed in your browser ?

Technical details
Node id
n-2ab5cee8137b2478 · revision 1 · JSON
Definition source
derived:obreschkoff_combination
Fingerprint
f6-efce8cd29f4f1fd0

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): yes

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: 4,  2;
  1: 4,  6,   2;
  2: 4, 10,   8,   2;
  3: 4, 14,  18,  10,   2;
  4: 4, 18,  32,  28,  12,   2;
  5: 4, 22,  50,  60,  40,  14,   2;
  6: 4, 26,  72, 110, 100,  54,  16,   2;
  7: 4, 30,  98, 182, 210, 154,  70,  18,   2;
  8: 4, 34, 128, 280, 392, 364, 224,  88,  20,  2;
  9: 4, 38, 162, 408, 672, 756, 588, 312, 108, 22, 2;
%F Formula
The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 2*x.
%t Mathematica
P[0, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 4 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + x)*v[i - 1]); 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-0787c477c17aca9a

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.