Real-rooted atlas

A172431 → D → D

nonnegative coefficients

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

Recurrence

\(\left(12 - 14\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(2 + 6\,x - 6\,n - 30\,n x + 4\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(x + 4\,x^{2} - 2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 6\)

Rows ?

nx0x1x2x3x4x5Pn(1)
00
10
266
3202444
44212060222
572336420120948
6110720151211202103672
7156132039605040252033613332

All 15 rows

Technical details
Node id
n-21cce4e621ee3f6c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b35c5c8db9d096e2

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 -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:   6;
  3:  20,   24;
  4:  42,  120,    60;
  5:  72,  336,   420,   120;
  6: 110,  720,  1512,  1120,   210;
  7: 156, 1320,  3960,  5040,  2520,   336;
  8: 210, 2184,  8580, 15840, 13860,  5040,  504;
  9: 272, 3360, 16380, 40040, 51480, 33264, 9240, 720;
%F Formula
The row polynomials satisfy (4*n^2-14*n+12)*P(n,x) = (4*n^2-6*n+2+(12*n^2-30*n+6)*x)*P(n-1,x) - ((2*n-1)*x+(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 6*n + 2 + (12*n^2 - 30*n + 6)*x)*P[n - 1, x] - ((2*n - 1)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12)]];
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] = 6; for(n = 3, N, my(i = n + 1); v[i] = ((4*n^2 - 6*n + 2 + (12*n^2 - 30*n + 6)*x)*v[i - 1] - ((2*n - 1)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12)); 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