Real-rooted atlas

A098158 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 20\,x + 3\,n - 18\,n x - n^{2} + 4\,n^{2} x\right)\,P_{n}(x) = \left(4 - 20\,x + 34\,x^{2} - 5\,n + 27\,n x - 46\,n x^{2} + n^{2} - 7\,n^{2} x + 12\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(2\,x - 10\,x^{2} + 8\,x^{3} - 2\,n x + 10\,n x^{2} - 8\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3123
40639
50218424
6001540560
700360756144
8000281751267336

All 15 rows

Technical details
Node id
n-bdbabe9e8ea4ed15 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-40fd595992edbfb8

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: 1;
   3: 1, 2;
   4: 0, 6,  3;
   5: 0, 2, 18,  4;
   6: 0, 0, 15, 40,   5;
   7: 0, 0,  3, 60,  75,   6;
   8: 0, 0,  0, 28, 175, 126,   7;
   9: 0, 0,  0,  4, 140, 420, 196,   8;
  10: 0, 0,  0,  0,  45, 504, 882, 288, 9;
%F Formula
The row polynomials satisfy -(n^2-3*n+2-(4*n^2-18*n+20)*x)*P(n,x) = (n^2-5*n+4-(7*n^2-27*n+20)*x+(12*n^2-46*n+34)*x^2)*P(n-1,x) - ((2*n-2)*x-(10*n-10)*x^2+(8*n-8)*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 - 5*n + 4 - (7*n^2 - 27*n + 20)*x + (12*n^2 - 46*n + 34)*x^2)*P[n - 1, x] - ((2*n - 2)*x - (10*n - 10)*x^2 + (8*n - 8)*x^3)*D[P[n - 1, x], x])/(-n^2 + 3*n - 2 + (4*n^2 - 18*n + 20)*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] = 1; for(n = 3, N, my(i = n); v[i] = ((n^2 - 5*n + 4 - (7*n^2 - 27*n + 20)*x + (12*n^2 - 46*n + 34)*x^2)*v[i - 1] - ((2*n - 2)*x - (10*n - 10)*x^2 + (8*n - 8)*x^3)*deriv(v[i - 1], x))/(-n^2 + 3*n - 2 + (4*n^2 - 18*n + 20)*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. [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.