Real-rooted atlas

A356546 → 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

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-144 + 96\,x + 384\,n - 256\,n x - 192\,n^{2} + 128\,n^{2} x\right)\,P_{n-2}(x) + \left(16 + 8\,x - 48\,n - 24\,n x + 32\,n^{2} + 16\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 12 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
112820
221638496696
34320131208320128027040
490720394240439040161280179201103200
51959552111121921838592011612160296755225804846295424
6431101443018301446743869446310993922715525125298585637847041978749696
7960740352800660889622729175040288775618561822551244858900807689277931525622988885673702400

All 15 rows

Technical details
Node id
n-f522163d1450b482 · 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:          216,           384,             96;
  3:         4320,         13120,           8320,           1280;
  4:        90720,        394240,         439040,         161280,          17920;
  5:      1959552,      11112192,       18385920,       11612160,        2967552,         258048;
  6:     43110144,     301830144,      674386944,      631099392,      271552512,       52985856,        3784704;
  7:    960740352,    8006608896,    22729175040,    28877561856,    18225512448,     5890080768,      927793152,      56229888;
  8:  21616657920,  208964321280,   722887925760,  1176926699520,  1004863242240,   471487610880,   121456558080,   16025518080,    843448320;
  9: 489977579520, 5389778268160, 22048418529280, 44159767674880, 48328323563520, 30498247802880, 11305206415360, 2415261122560, 274026987520, 12745441280;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = -(192*n^2-384*n+144-(128*n^2-256*n+96)*x)*P(n-2,x) + (32*n^2-48*n+16+(16*n^2-24*n+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[Together[(-(192*n^2 - 384*n + 144 - (128*n^2 - 256*n + 96)*x)*P[n - 2, x] + (32*n^2 - 48*n + 16 + (16*n^2 - 24*n + 8)*x)*P[n - 1, x])/(n^2 - n)]];
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] = (-(192*n^2 - 384*n + 144 - (128*n^2 - 256*n + 96)*x)*v[i - 2] + (32*n^2 - 48*n + 16 + (16*n^2 - 24*n + 8)*x)*v[i - 1])/(n^2 - n)); 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.