Real-rooted atlas

A028326 → basis A013611 → basis A008288

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-12 + 8\,x\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 12 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
112820
27212832232
343213128321282704
425921126412544460851231520
5155528819214592092160235522048367424
6933126533121459712136601658777611468881924283008
7559872466585613245440168284161062092834324485406723276849926400

All 15 rows

Technical details
Node id
n-b8f00920709d45e5 · 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:        2;
  1:       12,         8;
  2:       72,       128,        32;
  3:      432,      1312,       832,        128;
  4:     2592,     11264,     12544,       4608,        512;
  5:    15552,     88192,    145920,      92160,      23552,       2048;
  6:    93312,    653312,   1459712,    1366016,     587776,     114688,      8192;
  7:   559872,   4665856,  13245440,   16828416,   10620928,    3432448,    540672,    32768;
  8:  3359232,  32473088, 112336896,  182894592,  156155904,   73269248,  18874368,  2490368,   131072;
  9: 20155392, 221710336, 906969088, 1816526848, 1988001792, 1254555648, 465043456, 99352576, 11272192, 524288;
%F Formula
The row polynomials satisfy P(n,x) = -(12-8*x)*P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 12 + 8*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 12 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(12 - 8*x)*P[n - 2, x] + (8 + 4*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] = 2; v[2] = 12 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(12 - 8*x)*v[i - 2] + (8 + 4*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.