Real-rooted atlas

A038208 → 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

\(P_{n}(x) = \left(-48 + 32\,x\right)\,P_{n-2}(x) + \left(16 + 8\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 12 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
112820
214425664464
317285248332851210816
42073690112100352368644096252160
5248832141107223347201474560376832327685878784
62985984209059844671078443712512188088323670016262144137056256
73583180829861478484770816010770186246797393922196766723460300820971523195289600

All 15 rows

Technical details
Node id
n-e21c48f32e7cdb91 · 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:          1;
  1:         12,           8;
  2:        144,         256,           64;
  3:       1728,        5248,         3328,          512;
  4:      20736,       90112,       100352,        36864,         4096;
  5:     248832,     1411072,      2334720,      1474560,       376832,        32768;
  6:    2985984,    20905984,     46710784,     43712512,     18808832,      3670016,       262144;
  7:   35831808,   298614784,    847708160,   1077018624,    679739392,    219676672,     34603008,     2097152;
  8:  429981696,  4156555264,  14379122688,  23410507776,  19987955712,   9378463744,   2415919104,   318767104,   16777216;
  9: 5159780352, 56757846016, 232184086528, 465030873088, 508928458752, 321166245888, 119051124736, 25434259456, 2885681152, 134217728;
%F Formula
The row polynomials satisfy P(n,x) = -(48-32*x)*P(n-2,x) + (16+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 12 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 12 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(48 - 32*x)*P[n - 2, x] + (16 + 8*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] = 12 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(48 - 32*x)*v[i - 2] + (16 + 8*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.