Real-rooted atlas

A026729 → θ+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 6\,x + x^{2} - x^{3} + 5\,n - 8\,n x - 2\,n x^{2} + n x^{3} - 4\,n^{2} + 2\,n^{2} x + n^{2} x^{2} + n^{3}\right)\,P_{n}(x) = \left(16 - 20\,x - 10\,x^{2} + 4\,x^{3} + 2\,x^{4} - 28\,n + 8\,n x + 8\,n x^{2} + 14\,n^{2} - 2\,n^{2} x^{2} - 2\,n^{3}\right)\,P_{n-1}(x) + \left(8\,x - 14\,x^{2} + 5\,x^{4} + x^{5} - 12\,n x + 5\,n x^{2} + 6\,n x^{3} + n x^{4} + 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
328313
401218434
5063932582
6003692506184
70012124180727395
800080320312988818

All 15 rows

Technical details
Node id
n-e621497990f477e0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a02c93eddb0130ce

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 -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: 2,  2;
   3: 2,  8,  3;
   4: 0, 12, 18,   4;
   5: 0,  6, 39,  32,   5;
   6: 0,  0, 36,  92,  50,   6;
   7: 0,  0, 12, 124, 180,  72,    7;
   8: 0,  0,  0,  80, 320, 312,   98,   8;
   9: 0,  0,  0,  20, 305, 690,  497, 128,   9;
  10: 0,  0,  0,   0, 150, 870, 1316, 744, 162, 10;
%F Formula
The row polynomials satisfy (n^3-4*n^2+5*n-2+(2*n^2-8*n+6)*x+(n^2-2*n+1)*x^2+(n-1)*x^3)*P(n,x) = -(2*n^3-14*n^2+28*n-16-(8*n-20)*x+(2*n^2-8*n+10)*x^2-4*x^3-2*x^4)*P(n-1,x) + ((4*n^2-12*n+8)*x+(n^2+5*n-14)*x^2+6*n*x^3+(n+5)*x^4+x^5)*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[(-(2*n^3 - 14*n^2 + 28*n - 16 - (8*n - 20)*x + (2*n^2 - 8*n + 10)*x^2 - 4*x^3 - 2*x^4)*P[n - 1, x] + ((4*n^2 - 12*n + 8)*x + (n^2 + 5*n - 14)*x^2 + 6*n*x^3 + (n + 5)*x^4 + x^5)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + 5*n - 2 + (2*n^2 - 8*n + 6)*x + (n^2 - 2*n + 1)*x^2 + (n - 1)*x^3)]];
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] = (-(2*n^3 - 14*n^2 + 28*n - 16 - (8*n - 20)*x + (2*n^2 - 8*n + 10)*x^2 - 4*x^3 - 2*x^4)*v[i - 1] + ((4*n^2 - 12*n + 8)*x + (n^2 + 5*n - 14)*x^2 + 6*n*x^3 + (n + 5)*x^4 + x^5)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + 5*n - 2 + (2*n^2 - 8*n + 6)*x + (n^2 - 2*n + 1)*x^2 + (n - 1)*x^3)); 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.