Real-rooted atlas

A099097 → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-27 + 3\,x + 27\,n - 6\,n x + 3\,n^{2} x\right)\,P_{n}(x) = \left(-81 - 63\,x - 4\,x^{2} + 81\,n + 18\,n x - 2\,n x^{2} + 9\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-81\,x - 45\,x^{2} - 4\,x^{3} - 9\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
233
39211
4271239
581543138
624321627486
772981016241705
821872916810485961
96561102063645360520777
1019683349921530921607572219
115904911809861236113406756250404
12177147393660236196544324725108866268
13531441129907888573524494428350113472990689
141594323425152832476951049760153090907214710305615
15478296913817466116917024330260765450612361764835450855
Technical details
Node id
n-8e2140e39437a3ed · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-eb52c46fd35a0a6d

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept -1)
  • 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:     1;
   2:     3;
   3:     9,     2;
   4:    27,    12;
   5:    81,    54,     3;
   6:   243,   216,    27;
   7:   729,   810,   162,    4;
   8:  2187,  2916,   810,   48;
   9:  6561, 10206,  3645,  360,  5;
  10: 19683, 34992, 15309, 2160, 75;
%F Formula
The row polynomials satisfy (27*n-27+(3*n^2-6*n+3)*x)*P(n,x) = (81*n-81+(9*n^2+18*n-63)*x+(2*n^2-2*n-4)*x^2)*P(n-1,x) - (81*x+(9*n+45)*x^2+(4*n+4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((81*n - 81 + (9*n^2 + 18*n - 63)*x + (2*n^2 - 2*n - 4)*x^2)*P[n - 1, x] - (81*x + (9*n + 45)*x^2 + (4*n + 4)*x^3)*D[P[n - 1, x], x])/(27*n - 27 + (3*n^2 - 6*n + 3)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((81*n - 81 + (9*n^2 + 18*n - 63)*x + (2*n^2 - 2*n - 4)*x^2)*v[i - 1] - (81*x + (9*n + 45)*x^2 + (4*n + 4)*x^3)*deriv(v[i - 1], x))/(27*n - 27 + (3*n^2 - 6*n + 3)*x)); 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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.