Real-rooted atlas

A181371: Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (0 <= k <= floor(n/2)).

A011973 → x ↦ (1/9)x−1/9

A181371 nonnegative coefficients

Proved for all n: all roots real, in (−∞, 1] what this means

Recurrence from OEIS A181371
\(P_{n}(x) = \left(3\right)\,P_{n-1}(x) + \left(-1 + x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3Pn(1)
011
133
2819
321627
45525181
5144909243
6377300511729
7987954234122187

Show all 15 stored rows

Generated b-files: A000244 (row sums), A181371 ?

Computed in your browser ?

Technical details
Node id
n-1fe6eede339bc6ea · revision 1 · JSON
Definition source
oeis_catalog:A181371
Fingerprint
f6-776041ebb0e099b1

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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 ?

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:    1;
  1:    3;
  2:    8,    1;
  3:   21,    6;
  4:   55,   25,    1;
  5:  144,   90,    9;
  6:  377,  300,   51,   1;
  7:  987,  954,  234,  12;
  8: 2584, 2939,  951,  86,  1;
  9: 6765, 8850, 3573, 480, 15;
%F Formula
The row polynomials satisfy P(n,x) = 3*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[3*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = 3*v[i - 1] - (1 - x)*v[i - 2]); 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.

A181371: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:    1;
  1:    3;
  2:    8,    1;
  3:   21,    6;
  4:   55,   25,    1;
  5:  144,   90,    9;
  6:  377,  300,   51,   1;
  7:  987,  954,  234,  12;
  8: 2584, 2939,  951,  86,  1;
  9: 6765, 8850, 3573, 480, 15;
%F Formula
The row polynomials satisfy P(n,x) = 3*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[3*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = 3*v[i - 1] - (1 - x)*v[i - 2]); 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, 1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-6b3e76cd98792eb8

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. [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