Real-rooted atlas

A125662 → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 20\,x - 18\,n - 30\,n x + 12\,n^{2} + 10\,n^{2} x\right)\,P_{n}(x) = \left(-4\,x + 30\,x^{2} + 50\,x^{3} - 6\,n - 78\,n x - 165\,n x^{2} - 75\,n x^{3} + 12\,n^{2} + 64\,n^{2} x + 75\,n^{2} x^{2} + 25\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(5\,x^{2} + 30\,x^{3} + 25\,x^{4} - 6\,n x - 41\,n x^{2} - 60\,n x^{3} - 25\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
261622
395063122
412102270220604
51517270212007202809
618260144038044770226212554
721366256592201786517634690954580

All 15 rows

Technical details
Node id
n-31299e3f88545fde · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e54c281d84b9100e

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 -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:  0;
  1:  3;
  2:  6,  16;
  3:  9,  50,   63;
  4: 12, 102,  270,   220;
  5: 15, 172,  702,  1200,    720;
  6: 18, 260, 1440,  3804,   4770,   2262;
  7: 21, 366, 2565,  9220,  17865,  17634,   6909;
  8: 24, 490, 4158, 18960,  50040,  76242,  61950,  20672;
  9: 27, 632, 6300, 34860, 116865, 242556, 303786, 209560, 60885;
%F Formula
The row polynomials satisfy (12*n^2-18*n+6+(10*n^2-30*n+20)*x)*P(n,x) = (12*n^2-6*n+(64*n^2-78*n-4)*x+(75*n^2-165*n+30)*x^2+(25*n^2-75*n+50)*x^3)*P(n-1,x) - (6*n*x+(41*n-5)*x^2+(60*n-30)*x^3+(25*n-25)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 6*n + (64*n^2 - 78*n - 4)*x + (75*n^2 - 165*n + 30)*x^2 + (25*n^2 - 75*n + 50)*x^3)*P[n - 1, x] - (6*n*x + (41*n - 5)*x^2 + (60*n - 30)*x^3 + (25*n - 25)*x^4)*D[P[n - 1, x], x])/(12*n^2 - 18*n + 6 + (10*n^2 - 30*n + 20)*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] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^2 - 6*n + (64*n^2 - 78*n - 4)*x + (75*n^2 - 165*n + 30)*x^2 + (25*n^2 - 75*n + 50)*x^3)*v[i - 1] - (6*n*x + (41*n - 5)*x^2 + (60*n - 30)*x^3 + (25*n - 25)*x^4)*deriv(v[i - 1], x))/(12*n^2 - 18*n + 6 + (10*n^2 - 30*n + 20)*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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  3. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023