Real-rooted atlas

A136572 → basis A013610 → basis A008288

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4\,n - 6\,n x - 4\,n^{2} + 6\,n^{2} x\right)\,P_{n-2}(x) + \left(5\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
2326618116
338413509721622868
46144307443952816200194494560
512288079884015336001101600311040291603897120
62949120235936806117768065823840318427206823440524880192735360
7825753607844709602580348960378996408027221443209810007201690113601102248011120538240

All 15 rows

Technical details
Node id
n-71c0b2a084368588 · revision 0 · JSON
Definition source
derived:basis_map_A008288
Fingerprint
f6-52999a0df2a941cc

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:           4,             3;
  2:          32,            66,            18;
  3:         384,          1350,           972,            162;
  4:        6144,         30744,         39528,          16200,           1944;
  5:      122880,        798840,       1533600,        1101600,         311040,          29160;
  6:     2949120,      23593680,      61177680,       65823840,       31842720,        6823440,        524880;
  7:    82575360,     784470960,    2580348960,     3789964080,     2722144320,      981000720,     169011360,      11022480;
  8:  2642411520,   29066567040,  116264937600,   219338098560,   214828951680,   113742195840,   32479574400,    4673531520,    264539520;
  9: 95126814720, 1189085546880, 5612470456320, 13032568846080, 16442718405120, 11812218647040, 4907737175040, 1157095860480, 142851340800, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n^2-4*n-(6*n^2-6*n)*x)*P(n-2,x) + (5*n+3*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 4*n - (6*n^2 - 6*n)*x)*P[n - 2, x] + (5*n + 3*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] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 4*n - (6*n^2 - 6*n)*x)*v[i - 2] + (5*n + 3*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.