Real-rooted atlas

A085881 → 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

\(P_{n}(x) = \left(-36 + 24\,x + 96\,n - 64\,n x - 48\,n^{2} + 32\,n^{2} x\right)\,P_{n-2}(x) + \left(-8 - 4\,x + 16\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
210819248348
332409840624096020280
4136080591360658560241920268801654800
5734832041670720689472004354560011128320967680173607840
648498912033955891207586853120709986816030549657605960908804257792022260934080
73782915136031526022528089496126720011370539980807176295526402319219302403653185536022140518403373402032000

All 15 rows

Technical details
Node id
n-e7fa785b1435b6e0 · 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:               6,                4;
  2:             108,              192,                48;
  3:            3240,             9840,              6240,               960;
  4:          136080,           591360,            658560,            241920,             26880;
  5:         7348320,         41670720,          68947200,          43545600,          11128320,            967680;
  6:       484989120,       3395589120,        7586853120,        7099868160,        3054965760,         596090880,         42577920;
  7:     37829151360,     315260225280,      894961267200,     1137053998080,      717629552640,      231921930240,      36531855360,       2214051840;
  8:   3404623622400,   32911880601600,   113854848307200,   185365955174400,   158265960652800,    74259298713600,   19129407897600,    2524019097600,    132843110400;
  9: 347271609484800, 3820005347558400, 15626816632627200, 31298235339571200, 34252699325644800, 21615633130291200, 8012565046886400, 1711816320614400, 194216627404800, 9033331507200;
%F Formula
The row polynomials satisfy P(n,x) = -(48*n^2-96*n+36-(32*n^2-64*n+24)*x)*P(n-2,x) + (16*n-8+(8*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(48*n^2 - 96*n + 36 - (32*n^2 - 64*n + 24)*x)*P[n - 2, x] + (16*n - 8 + (8*n - 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] = 1; v[2] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(48*n^2 - 96*n + 36 - (32*n^2 - 64*n + 24)*x)*v[i - 2] + (16*n - 8 + (8*n - 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.