Real-rooted atlas

A196347 → 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

\(P_{n}(x) = \left(12\,n - 8\,n x - 12\,n^{2} + 8\,n^{2} x\right)\,P_{n-2}(x) + \left(8\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16410
27212832232
31296393624963848112
431104135168150528552966144378240
5933120529152087552005529600141312012288022045440
6335923202351923205254963204917657602115993604128768029491201541882880
71410877440117579571203337850880042407608320267647385608649768960136249344082575360125814528000

All 15 rows

Technical details
Node id
n-0db0a2cf3d5aaadd · 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:            72,            128,              32;
  3:          1296,           3936,            2496,             384;
  4:         31104,         135168,          150528,           55296,            6144;
  5:        933120,        5291520,         8755200,         5529600,         1413120,          122880;
  6:      33592320,      235192320,       525496320,       491765760,       211599360,        41287680,        2949120;
  7:    1410877440,    11757957120,     33378508800,     42407608320,     26764738560,      8649768960,     1362493440,       82575360;
  8:   67722117120,   654657454080,   2264711823360,   3687154974720,   3148103024640,   1477108039680,   380507258880,    50205818880,    2642411520;
  9: 3656994324480, 40227123363840, 164560471326720, 329590631301120, 360703045140480, 227626576773120, 84377484656640, 18026531389440, 2045226516480, 95126814720;
%F Formula
The row polynomials satisfy P(n,x) = -(12*n^2-12*n-(8*n^2-8*n)*x)*P(n-2,x) + (8*n+4*n*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[-(12*n^2 - 12*n - (8*n^2 - 8*n)*x)*P[n - 2, x] + (8*n + 4*n*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] = -(12*n^2 - 12*n - (8*n^2 - 8*n)*x)*v[i - 2] + (8*n + 4*n*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.