Real-rooted atlas

reversed rows → D+1

nonnegative coefficients

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

Recurrence

\(\left(-n - 5\,n x + 4\,n x^{2} + 4\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-2\,n - 9\,n x + 2\,n x^{2} + 8\,n x^{3} + n^{2} + 4\,n^{2} x + 8\,n^{2} x^{2} + n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-3\,x^{2} - 16\,x^{3} - 16\,x^{4} - 2\,n x - 12\,n x^{2} - 16\,n x^{3} - n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
1325
2437
358215
4615526
572415248
683535785
794868242151

All 15 rows

Technical details
Node id
n-31ba686292905fdf · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4d17291db27785d7

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:  3,  2;
  2:  4,  3;
  3:  5,  8,   2;
  4:  6, 15,   5;
  5:  7, 24,  15,   2;
  6:  8, 35,  35,   7;
  7:  9, 48,  68,  24,  2;
  8: 10, 63, 117,  66,  9;
  9: 11, 80, 185, 150, 35, 2;
%F Formula
The row polynomials satisfy (n^3-n+(4*n^2-5*n)*x+4*n*x^2)*P(n,x) = (n^3+n^2-2*n+(2*n^3+4*n^2-9*n)*x+(8*n^2+2*n)*x^2+8*n*x^3)*P(n-1,x) - ((n^2+2*n)*x+(4*n^2+12*n+3)*x^2+(16*n+16)*x^3+16*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[((n^3 + n^2 - 2*n + (2*n^3 + 4*n^2 - 9*n)*x + (8*n^2 + 2*n)*x^2 + 8*n*x^3)*P[n - 1, x] - ((n^2 + 2*n)*x + (4*n^2 + 12*n + 3)*x^2 + (16*n + 16)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^3 - n + (4*n^2 - 5*n)*x + 4*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] = ((n^3 + n^2 - 2*n + (2*n^3 + 4*n^2 - 9*n)*x + (8*n^2 + 2*n)*x^2 + 8*n*x^3)*v[i - 1] - ((n^2 + 2*n)*x + (4*n^2 + 12*n + 3)*x^2 + (16*n + 16)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^3 - n + (4*n^2 - 5*n)*x + 4*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

Same fingerprint ?

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.