Real-rooted atlas

A111006 → basis A008288

nonnegative coefficients

Proved for all n all roots real, in [−3−2√2, −3+2√2]coefficients of one signHadamard kernel

Recurrence

\(P_{n}(x) = \left(2\,x\right)\,P_{n-4}(x) + \left(3\,x + x^{2}\right)\,P_{n-3}(x) + \left(2 + 6\,x + x^{2}\right)\,P_{n-2}(x) + \left(1 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)
\(P_{2}(x) = 3 + 7\,x + 2\,x^{2}\)
\(P_{3}(x) = 5 + 21\,x + 17\,x^{2} + 3\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
237212
352117346
4116391405210
521157345273828886
64339111551383722163133870
78591734415681446917213112116646

All 15 rows

Technical details
Node id
n-ff2c124f57c19690 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-ce6d9281d03644ef

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   1,    1;
  2:   3,    7,     2;
  3:   5,   21,    17,     3;
  4:  11,   63,    91,    40,     5;
  5:  21,  157,   345,   273,    82,     8;
  6:  43,  391,  1155,  1383,   722,   163,    13;
  7:  85,  917,  3441,  5681,  4469,  1721,   311,   21;
  8: 171, 2127,  9675, 20583, 22291, 12760,  3864,  581,   34;
  9: 341, 4813, 25801, 67729, 95189, 74701, 33268, 8276, 1065, 55;
%F Formula
The row polynomials satisfy P(n,x) = 2*x*P(n-4,x) + (3*x+x^2)*P(n-3,x) + (2+6*x+x^2)*P(n-2,x) + (1+x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + x, P(2,x) = 3 + 7*x + 2*x^2, P(3,x) = 5 + 21*x + 17*x^2 + 3*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[2, x_] := 3 + 7*x + 2*x^2;
P[3, x_] := 5 + 21*x + 17*x^2 + 3*x^3;
P[n_, x_] := P[n, x] = Expand[2*x*P[n - 4, x] + (3*x + x^2)*P[n - 3, x] + (2 + 6*x + x^2)*P[n - 2, x] + (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] = 1; v[2] = 1 + x; v[3] = 3 + 7*x + 2*x^2; v[4] = 5 + 21*x + 17*x^2 + 3*x^3; for(n = 4, N, my(i = n + 1); v[i] = 2*x*v[i - 4] + (3*x + x^2)*v[i - 3] + (2 + 6*x + x^2)*v[i - 2] + (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)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [−3−2√2, −3+2√2] (rational outer bound [-1564556493/268435456, -46056243/268435456])
    by Affine-current basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

Proof chains ? JSON

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