Real-rooted atlas

reversed rows → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n^{2} + 3\,n^{2} x + n^{3} x\right)\,P_{n}(x) = \left(n + 6\,n x + 12\,n x^{2} + n^{2} + 7\,n^{2} x + 10\,n^{2} x^{2} + n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-x - 10\,x^{2} - 24\,x^{3} - n x - 9\,n x^{2} - 20\,n x^{3} - n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
1145
2167
318615
41101526
511227848
6114422885
7116606410151

All 15 rows

Technical details
Node id
n-3ce6048e351c7c7a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-11372483dbfbeb3e

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  4;
  2: 1,  6;
  3: 1,  8,   6;
  4: 1, 10,  15;
  5: 1, 12,  27,   8;
  6: 1, 14,  42,  28;
  7: 1, 16,  60,  64,  10;
  8: 1, 18,  81, 120,  45;
  9: 1, 20, 105, 200, 125, 12;
%F Formula
The row polynomials satisfy (n^2+n+(n^3+3*n^2+2*n)*x)*P(n,x) = (n^2+n+(n^3+7*n^2+6*n)*x+(2*n^3+10*n^2+12*n)*x^2)*P(n-1,x) - ((n+1)*x+(n^2+9*n+10)*x^2+(4*n^2+20*n+24)*x^3)*d/dx 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[Together[((n^2 + n + (n^3 + 7*n^2 + 6*n)*x + (2*n^3 + 10*n^2 + 12*n)*x^2)*P[n - 1, x] - ((n + 1)*x + (n^2 + 9*n + 10)*x^2 + (4*n^2 + 20*n + 24)*x^3)*D[P[n - 1, x], x])/(n^2 + n + (n^3 + 3*n^2 + 2*n)*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] = ((n^2 + n + (n^3 + 7*n^2 + 6*n)*x + (2*n^3 + 10*n^2 + 12*n)*x^2)*v[i - 1] - ((n + 1)*x + (n^2 + 9*n + 10)*x^2 + (4*n^2 + 20*n + 24)*x^3)*deriv(v[i - 1], x))/(n^2 + n + (n^3 + 3*n^2 + 2*n)*x)); 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. [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.