Real-rooted atlas

A209599 → D

nonnegative coefficients

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

Recurrence

\(\left(-14 - 4\,x + 9\,n - n^{2} + n^{2} x\right)\,P_{n}(x) = \left(-30 - 24\,x + 23\,n + 10\,n x - 6\,n x^{2} - 3\,n^{2} + n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-15 - 27\,x - 12\,x^{2} + 5\,n + 4\,n x - 5\,n x^{2} - 4\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
211
333
4729
515823
63024357
7586215135
8109146544313
920132416224711
1036568843510051593
1165514141083340353527
12116428322550102016567737
1320525556575428046254816839
143593107161255272242065252736409
Technical details
Node id
n-91dbf3ed4f4cdb7e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-22048d402d1d57cf

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   0;
  2:   1;
  3:   3;
  4:   7,   2;
  5:  15,   8;
  6:  30,  24,   3;
  7:  58,  62,  15;
  8: 109, 146,  54,  4;
  9: 201, 324, 162, 24;
%F Formula
The row polynomials satisfy -(n^2-9*n+14-(n^2-4)*x)*P(n,x) = -(3*n^2-23*n+30-(n^2+10*n-24)*x-(2*n^2-6*n)*x^2)*P(n-1,x) + (5*n-15+(4*n-27)*x-(5*n+12)*x^2-4*n*x^3)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n^2 - 23*n + 30 - (n^2 + 10*n - 24)*x - (2*n^2 - 6*n)*x^2)*P[n - 1, x] + (5*n - 15 + (4*n - 27)*x - (5*n + 12)*x^2 - 4*n*x^3)*D[P[n - 1, x], x])/(-n^2 + 9*n - 14 + (n^2 - 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(3*n^2 - 23*n + 30 - (n^2 + 10*n - 24)*x - (2*n^2 - 6*n)*x^2)*v[i - 1] + (5*n - 15 + (4*n - 27)*x - (5*n + 12)*x^2 - 4*n*x^3)*deriv(v[i - 1], x))/(-n^2 + 9*n - 14 + (n^2 - 4)*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

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