Real-rooted atlas

A120909 → basis A013610 → basis A008288

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-27 + 18\,x\right)\,P_{n-2}(x) + \left(12 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 3\)
\(P_{1}(x) = 27 + 18\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
033
1271845
2243432108783
321876642421264813689
4196838553695256349923888239355
5177147100456216621201049760268272233284185189
6159432311162448249405482333966410042704195955213996873179207
71434890711958078633946614043129389627220276887969888138568328398081279559025

All 15 rows

Technical details
Node id
n-1a789ecfc71669a3 · 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 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:          3;
  1:         27,          18;
  2:        243,         432,         108;
  3:       2187,        6642,        4212,          648;
  4:      19683,       85536,       95256,        34992,         3888;
  5:     177147,     1004562,     1662120,      1049760,       268272,       23328;
  6:    1594323,    11162448,    24940548,     23339664,     10042704,     1959552,      139968;
  7:   14348907,   119580786,   339466140,    431293896,    272202768,    87969888,    13856832,     839808;
  8:  129140163,  1248374592,  4318607664,   7031082528,   6003157536,  2816716032,   725594112,   95738112,   5038848;
  9: 1162261467, 12784935186, 52300407888, 104750091648, 114638200992, 72344000448, 26816749056, 5729170176, 650011392, 30233088;
%F Formula
The row polynomials satisfy P(n,x) = -(27-18*x)*P(n-2,x) + (12+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 3, P(1,x) = 27 + 18*x.
%t Mathematica
P[0, x_] := 3;
P[1, x_] := 27 + 18*x;
P[n_, x_] := P[n, x] = Expand[-(27 - 18*x)*P[n - 2, x] + (12 + 6*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] = 3; v[2] = 27 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = -(27 - 18*x)*v[i - 2] + (12 + 6*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

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.