Real-rooted atlas

A111125 → D → D

nonnegative coefficients

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

Recurrence

\(\left(9 - 9\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(-1 - x + n + 2\,n^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(-4\,x - x^{2} + 4\,n x + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 2\)

Rows ?

nx0x1x2x3x4x5Pn(1)
10
20
322
414620
5545412120
615426413220570
7364936780260302370
8756270033001800450429048

All 15 rows

Technical details
Node id
n-5cf9c2bd32c3fc06 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e7fa2f49aa742843

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -3)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    0;
   2:    0;
   3:    2;
   4:   14,     6;
   5:   54,    54,    12;
   6:  154,   264,   132,    20;
   7:  364,   936,   780,   260,    30;
   8:  756,  2700,  3300,  1800,   450,   42;
   9: 1428,  6732, 11220,  8840,  3570,  714,   56;
  10: 2508, 15048, 32604, 34580, 19950, 6384, 1064, 72;
%F Formula
The row polynomials satisfy (2*n^2-9*n+9)*P(n,x) = (2*n^2+n-1+(n^2-1)*x)*P(n-1,x) + ((4*n-4)*x+(n-1)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n - 1 + (n^2 - 1)*x)*P[n - 1, x] + ((4*n - 4)*x + (n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 9*n + 9)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 2; for(n = 4, N, my(i = n); v[i] = ((2*n^2 + n - 1 + (n^2 - 1)*x)*v[i - 1] + ((4*n - 4)*x + (n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 9*n + 9)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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