Real-rooted atlas

A055151 → D → D

nonnegative coefficients

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

Recurrence

\(\left(-8 - 2\,n + n^{2}\right)\,P_{n}(x) = \left(2\,n - 20\,n x + n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 0\)
\(P_{4}(x) = 4\)

Rows ?

nx0x1x2x3x4x5Pn(1)
00
10
20
30
444
52020
6603090
7140210350
82808401681288
9504252015124536
108406300756084015540
1113201386027720924052140
1219802772083160554403960172260
1328605148021621624024051480562276
14400490090504504840840360360180181817816
Technical details
Node id
n-2f1d141549840af4 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5b6b8f74397d2d43

Observed on the stored rows ?

  • Degrees: -1, -1, -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5 (half affine: slope 1, intercept -4)
  • 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:   0;
  3:   0;
  4:   4;
  5:  20;
  6:  60,   30;
  7: 140,  210;
  8: 280,  840,  168;
  9: 504, 2520, 1512;
%F Formula
The row polynomials satisfy (n^2-2*n-8)*P(n,x) = (n^2+2*n+(4*n^2-20*n)*x)*P(n-1,x) + (2*n*x-8*n*x^2)*d/dx P(n-1,x) for n >= 5, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 0, P(4,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 0;
P[4, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n + (4*n^2 - 20*n)*x)*P[n - 1, x] + (2*n*x - 8*n*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n - 8)]];
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] = 0; v[4] = 0; v[5] = 4; for(n = 5, N, my(i = n + 1); v[i] = ((n^2 + 2*n + (4*n^2 - 20*n)*x)*v[i - 1] + (2*n*x - 8*n*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n - 8)); 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. [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