Real-rooted atlas

A059364 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
218331566
31624233661051056
4194465348199455494522176
5291601193941948591584996426910395576576
65248802528172505958453861493217329102260713513517873856
7110224806096481214417384418900288914835614469734682181769402027025643458816

All 15 rows

Technical details
Node id
n-40fdee53f6f3666d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-f8d027db0af7f231

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:          3,           3;
  2:         18,          33,           15;
  3:        162,         423,          366,          105;
  4:       1944,        6534,         8199,         4554,          945;
  5:      29160,      119394,       194859,       158499,        64269,        10395;
  6:     524880,     2528172,      5059584,      5386149,      3217329,      1022607,       135135;
  7:   11022480,    60964812,    144173844,    189002889,    148356144,     69734682,     18176940,     2027025;
  8:  264539520,  1650537648,   4496574060,   6987024684,   6773596569,   4195686816,   1621736154,   357656580,   34459425;
  9: 7142567040, 49590767376, 152767714932, 274084573608, 315640576359, 241981878843, 123504925662, 40469714586, 7725879495, 654729075;
%F Formula
The row polynomials satisfy P(n,x) = (3*n+(2*n+1)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(3*n + (2*n + 1)*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; for(n = 1, N, my(i = n + 1); v[i] = (3*n + (2*n + 1)*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

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.