Real-rooted atlas

A026729 → θ+1 → D

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x - 3\,n - 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-16 - 20\,x - 6\,x^{2} + 12\,n + 8\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 10\,x^{2} - 2\,x^{3} + 4\,n x + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
222
3268
40121224
506362062
600368030146
7001212015042324
80008030025256688

All 15 rows

Technical details
Node id
n-c34001663e229dee · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-81307b53201494f8

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: 2;
   3: 2,  6;
   4: 0, 12, 12;
   5: 0,  6, 36,  20;
   6: 0,  0, 36,  80,  30;
   7: 0,  0, 12, 120, 150,  42;
   8: 0,  0,  0,  80, 300, 252,   56;
   9: 0,  0,  0,  20, 300, 630,  392,  72;
  10: 0,  0,  0,   0, 150, 840, 1176, 576, 90;
%F Formula
The row polynomials satisfy (n^2-3*n+2-(2*n-4)*x)*P(n,x) = -(2*n^2-12*n+16-(8*n-20)*x+6*x^2)*P(n-1,x) + ((4*n-8)*x+(n-10)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 12*n + 16 - (8*n - 20)*x + 6*x^2)*P[n - 1, x] + ((4*n - 8)*x + (n - 10)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n^2 - 3*n + 2 - (2*n - 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 12*n + 16 - (8*n - 20)*x + 6*x^2)*v[i - 1] + ((4*n - 8)*x + (n - 10)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n^2 - 3*n + 2 - (2*n - 4)*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.