Real-rooted atlas

A128099 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-8\,n - 3\,n x + 8\,n x^{2} + 3\,n^{2} + 8\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(12 - 8\,x - 52\,x^{2} - 32\,x^{3} - 34\,n - 39\,n x + 28\,n x^{2} + 32\,n x^{3} + 9\,n^{2} + 44\,n^{2} x + 32\,n^{2} x^{2} + 10\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(-45 - 202\,x - 333\,x^{2} - 240\,x^{3} - 64\,x^{4} - 63\,n - 191\,n x - 192\,n x^{2} - 64\,n x^{3} - 18\,n^{2} - 34\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2527
39413
42522451
5535612121
6125178728383
727346023232997

All 15 rows

Technical details
Node id
n-4c3137a359e68165 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-500426ab9288f69f

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 0)
  • 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;
  2:    5,    2;
  3:    9,    4;
  4:   25,   22,    4;
  5:   53,   56,   12;
  6:  125,  178,   72,   8;
  7:  273,  460,  232,  32;
  8:  609, 1230,  828, 208, 16;
  9: 1325, 3072, 2484, 800, 80;
%F Formula
The row polynomials satisfy (2*n^3+3*n^2-8*n+(8*n^2-3*n)*x+8*n*x^2)*P(n,x) = (10*n^3+9*n^2-34*n+12+(8*n^3+44*n^2-39*n-8)*x+(32*n^2+28*n-52)*x^2+(32*n-32)*x^3)*P(n-1,x) - (18*n^2+63*n+45+(34*n^2+191*n+202)*x+(16*n^2+192*n+333)*x^2+(64*n+240)*x^3+64*x^4)*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[((10*n^3 + 9*n^2 - 34*n + 12 + (8*n^3 + 44*n^2 - 39*n - 8)*x + (32*n^2 + 28*n - 52)*x^2 + (32*n - 32)*x^3)*P[n - 1, x] - (18*n^2 + 63*n + 45 + (34*n^2 + 191*n + 202)*x + (16*n^2 + 192*n + 333)*x^2 + (64*n + 240)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(2*n^3 + 3*n^2 - 8*n + (8*n^2 - 3*n)*x + 8*n*x^2)]];
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] = ((10*n^3 + 9*n^2 - 34*n + 12 + (8*n^3 + 44*n^2 - 39*n - 8)*x + (32*n^2 + 28*n - 52)*x^2 + (32*n - 32)*x^3)*v[i - 1] - (18*n^2 + 63*n + 45 + (34*n^2 + 191*n + 202)*x + (16*n^2 + 192*n + 333)*x^2 + (64*n + 240)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(2*n^3 + 3*n^2 - 8*n + (8*n^2 - 3*n)*x + 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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [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
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.