Real-rooted atlas

basis-map coefficients of A026729, shifted → basis A013609 → basis A008288

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

Recurrence

\(P_{n}(x) = \left(-3 + 2\,x\right)\,P_{n-2}(x) + \left(-4 - 2\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = -3 - 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2-3-2-5
3916429
4-27-82-52-8-169
58135239214416985
6-243-1378-2280-1440-368-32-5741
77295104114041067245928966433461
8-2187-18226-51740-65736-41488-13408-2112-128-195025
96561634242194083572163049921431043686448642561136689
10-19683-216514-885712-1773952-1941408-1225152-454144-97024-11008-512-6625109
1159049728272344450082344001111299289640964442496135552024704024576102438613965
12-177147-2421010-13010436-36276168-58644448-58289728-36786048-14924288-3860224-612864-54272-2048-225058681
135314417971616480068081533113442902599523457815042683243521380876804725939210592256148889611878440961311738121
14-1594323-26040610-173781176-626451360-1364281488-1906065632-1771081728-1117798656-483480832-142769664-28203008-3555328-258048-8192-7645370045
15478296984499120619128732248944737661458715049896000896107830481288135743744430391782416007823364150691847324876883681285570561638444560482149
Technical details
Node id
n-22d90d08a08421af · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-02d3ee6161499280

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 -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:     -3,      -2;
   3:      9,      16,       4;
   4:    -27,     -82,     -52,       -8;
   5:     81,     352,     392,      144,       16;
   6:   -243,   -1378,   -2280,    -1440,     -368,      -32;
   7:    729,    5104,   11404,    10672,     4592,      896,      64;
   8:  -2187,  -18226,  -51740,   -65736,   -41488,   -13408,   -2112,   -128;
   9:   6561,   63424,  219408,   357216,   304992,   143104,   36864,   4864,    256;
  10: -19683, -216514, -885712, -1773952, -1941408, -1225152, -454144, -97024, -11008, -512;
%F Formula
The row polynomials satisfy P(n,x) = -(3-2*x)*P(n-2,x) - (4+2*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = -3 - 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := -3 - 2*x;
P[n_, x_] := P[n, x] = Expand[-(3 - 2*x)*P[n - 2, x] - (4 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = -3 - 2*x; for(n = 3, N, my(i = n); v[i] = -(3 - 2*x)*v[i - 2] - (4 + 2*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, v[i])))};
T(10)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
    derived from Affine-current basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Affine-current basis map theorem

Proof chains ? JSON

Same fingerprint ?