Real-rooted atlas

A055151 → D → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-4 + n^{2}\right)\,P_{n}(x) = \left(-10\,n - 12\,n x + 5\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,n - 14\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
211
333
410414
5302050
6909015195
7266350105721
87841288588562716
923044536277250410116
1067651554011970336021037845
11198555214048510184802310141295
12582781722601881009108017820792528330
13171106562276705276415272111540102961975766
145025931817816257657417897886156159009030037395479
Technical details
Node id
n-f4289e05baefa106 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-43802ce87d578bbc

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:   10,    4;
  5:   30,   20;
  6:   90,   90,   15;
  7:  266,  350,  105;
  8:  784, 1288,  588,  56;
  9: 2304, 4536, 2772, 504;
%F Formula
The row polynomials satisfy (n^2-4)*P(n,x) = (5*n^2-10*n+(4*n^2-12*n)*x)*P(n-1,x) - (6*n+14*n*x+8*n*x^2)*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[((5*n^2 - 10*n + (4*n^2 - 12*n)*x)*P[n - 1, x] - (6*n + 14*n*x + 8*n*x^2)*D[P[n - 1, x], x])/(n^2 - 4)]];
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] = ((5*n^2 - 10*n + (4*n^2 - 12*n)*x)*v[i - 1] - (6*n + 14*n*x + 8*n*x^2)*deriv(v[i - 1], x))/(n^2 - 4)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023