Real-rooted atlas

A165253 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 12\,x + 3\,x^{2} + 11\,n - 8\,n x - 2\,n x^{2} - 9\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(9 + 23\,x + 9\,x^{2} + x^{3} - 4\,n - 18\,n x - 8\,n x^{2} - n x^{3} - 7\,n^{2} - 2\,n^{2} x + 3\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-30 - 66\,x - 47\,x^{2} - 12\,x^{3} - x^{4} - 10\,n - 12\,n x - 2\,n x^{2} + 5\,n^{2} + 6\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2314
3107118
4323511179
59914975151339
62995764051301911430
7887208818998542002315952

All 15 rows

Technical details
Node id
n-a2ba570e434d7f6e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-6af2fb3d8a3643c2

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:    1;
  1:    1;
  2:    3,     1;
  3:   10,     7,     1;
  4:   32,    35,    11,     1;
  5:   99,   149,    75,    15,    1;
  6:  299,   576,   405,   130,   19,    1;
  7:  887,  2088,  1899,   854,  200,   23,   1;
  8: 2595,  7229,  8097,  4725, 1550,  285,  27,  1;
  9: 7508, 24179, 32243, 23243, 9900, 2547, 385, 31, 1;
%F Formula
The row polynomials satisfy (2*n^3-9*n^2+11*n-3-(8*n-12)*x-(2*n-3)*x^2)*P(n,x) = (3*n^3-7*n^2-4*n+9+(n^3-2*n^2-18*n+23)*x-(8*n-9)*x^2-(n-1)*x^3)*P(n-1,x) + (5*n^2-10*n-30+(6*n^2-12*n-66)*x+(n^2-2*n-47)*x^2-12*x^3-x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 7*n^2 - 4*n + 9 + (n^3 - 2*n^2 - 18*n + 23)*x - (8*n - 9)*x^2 - (n - 1)*x^3)*P[n - 1, x] + (5*n^2 - 10*n - 30 + (6*n^2 - 12*n - 66)*x + (n^2 - 2*n - 47)*x^2 - 12*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 - 9*n^2 + 11*n - 3 - (8*n - 12)*x - (2*n - 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((3*n^3 - 7*n^2 - 4*n + 9 + (n^3 - 2*n^2 - 18*n + 23)*x - (8*n - 9)*x^2 - (n - 1)*x^3)*v[i - 1] + (5*n^2 - 10*n - 30 + (6*n^2 - 12*n - 66)*x + (n^2 - 2*n - 47)*x^2 - 12*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 - 9*n^2 + 11*n - 3 - (8*n - 12)*x - (2*n - 3)*x^2)); 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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023