Real-rooted atlas

A085880 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 14\,n x - 8\,n x^{2} + n^{2} - 13\,n^{2} x + 12\,n^{2} x^{2} + 3\,n^{3} x - 6\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n}(x) = \left(6 - 18\,x - 18\,x^{2} + 6\,x^{3} - 10\,n + 48\,n x + 36\,n x^{2} - 22\,n x^{3} + 4\,n^{2} - 42\,n^{2} x - 16\,n^{2} x^{2} + 30\,n^{2} x^{3} + 12\,n^{3} x - 6\,n^{3} x^{2} - 18\,n^{3} x^{3} + 4\,n^{4} x^{2} + 4\,n^{4} x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
341620
415120135270
55667215128963136
621033601134013440525033600
779215840712801267209900028512342144
830037207240540596096011261256486481471473363360

All 15 rows

Technical details
Node id
n-d47e285bc2e91957 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     0;
   2:     1;
   3:     4,      16;
   4:    15,     120,      135;
   5:    56,     672,     1512,      896;
   6:   210,    3360,    11340,    13440,     5250;
   7:   792,   15840,    71280,   126720,    99000,    28512;
   8:  3003,   72072,   405405,   960960,  1126125,   648648,   147147;
   9: 11440,  320320,  2162160,  6406400, 10010000,  8648640,  3923920,   732160;
  10: 43758, 1400256, 11027016, 39207168, 76576500, 88216128, 60035976, 22404096, 3544398;
%F Formula
The row polynomials satisfy (n^2-2*n+(3*n^3-13*n^2+14*n)*x+(n^4-6*n^3+12*n^2-8*n)*x^2)*P(n,x) = (4*n^2-10*n+6+(12*n^3-42*n^2+48*n-18)*x+(4*n^4-6*n^3-16*n^2+36*n-18)*x^2+(4*n^4-18*n^3+30*n^2-22*n+6)*x^3)*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 10*n + 6 + (12*n^3 - 42*n^2 + 48*n - 18)*x + (4*n^4 - 6*n^3 - 16*n^2 + 36*n - 18)*x^2 + (4*n^4 - 18*n^3 + 30*n^2 - 22*n + 6)*x^3)*P[n - 1, x])/(n^2 - 2*n + (3*n^3 - 13*n^2 + 14*n)*x + (n^4 - 6*n^3 + 12*n^2 - 8*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 10*n + 6 + (12*n^3 - 42*n^2 + 48*n - 18)*x + (4*n^4 - 6*n^3 - 16*n^2 + 36*n - 18)*x^2 + (4*n^4 - 18*n^3 + 30*n^2 - 22*n + 6)*x^3)*v[i - 1])/(n^2 - 2*n + (3*n^3 - 13*n^2 + 14*n)*x + (n^4 - 6*n^3 + 12*n^2 - 8*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.