Real-rooted atlas

A055151 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-4 - 8\,x + 12\,n x + n^{2} - 2\,n^{2} x - 3\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(2\,n - 20\,n x + n^{2} + 4\,n^{2} x + 12\,n^{2} x^{2} - n^{3} x - 16\,n^{3} x^{2} + n^{4} x + 4\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(2\,n x - 8\,n x^{2} - 2\,n^{2} x^{2} + 8\,n^{2} x^{3} + 2\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
211
333
46814
5104050
61512045180
721280315616
82856012602242072
9361008378020166840
10451680945010080105022305
1155264020790369601155071995
1266396041580110880693004752230538
137857207722028828830030061776733382
149180081351356726721051050432432210212320409
Technical details
Node id
n-af6284f918f2b9ba · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8a27ad21a90fe89f

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:  6,    8;
  5: 10,   40;
  6: 15,  120,   45;
  7: 21,  280,  315;
  8: 28,  560, 1260,  224;
  9: 36, 1008, 3780, 2016;
%F Formula
The row polynomials satisfy (n^2-4+(n^4-3*n^3-2*n^2+12*n-8)*x)*P(n,x) = (n^2+2*n+(n^4-n^3+4*n^2-20*n)*x+(4*n^4-16*n^3+12*n^2)*x^2)*P(n-1,x) + (2*n*x+(2*n^3-2*n^2-8*n)*x^2-(8*n^3-8*n^2)*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[((n^2 + 2*n + (n^4 - n^3 + 4*n^2 - 20*n)*x + (4*n^4 - 16*n^3 + 12*n^2)*x^2)*P[n - 1, x] + (2*n*x + (2*n^3 - 2*n^2 - 8*n)*x^2 - (8*n^3 - 8*n^2)*x^3)*D[P[n - 1, x], x])/(n^2 - 4 + (n^4 - 3*n^3 - 2*n^2 + 12*n - 8)*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] = ((n^2 + 2*n + (n^4 - n^3 + 4*n^2 - 20*n)*x + (4*n^4 - 16*n^3 + 12*n^2)*x^2)*v[i - 1] + (2*n*x + (2*n^3 - 2*n^2 - 8*n)*x^2 - (8*n^3 - 8*n^2)*x^3)*deriv(v[i - 1], x))/(n^2 - 4 + (n^4 - 3*n^3 - 2*n^2 + 12*n - 8)*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. [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