Real-rooted atlas

A046716 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(18 + 12\,x + 2\,x^{2} - 9\,n - 6\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(-3 + 3\,n + n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
3915529
427997516217
581756963432652297
62437614137979486288332634349
772910230323522421521797596219391957674965
82187173429147375285346729319129210631791875751370016276481

All 15 rows

Technical details
Node id
n-38787ba8d451c816 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-86c6d94ba216abaa

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • 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:     1;
   2:     3,         2;
   3:     9,        15,          5;
   4:    27,        99,         75,         16;
   5:    81,       756,        963,        432,         65;
   6:   243,      7614,      13797,       9486,       2883,        326;
   7:   729,    102303,     235224,     215217,      97596,      21939,      1957;
   8:  2187,   1734291,    4737528,    5346729,    3191292,    1063179,    187575,     13700;
   9:  6561,  35166960,  110188350,  146805048,  107877339,   47250216,  12342510,   1781208,   109601;
  10: 19683, 824599602, 2902389570, 4439379636, 3856422393, 2082024486, 715681602, 153022284, 18613107, 986410;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n-18+(6*n-12)*x+(n-2)*x^2)*P(n-2,x) + (3*n-3+n*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(9*n - 18 + (6*n - 12)*x + (n - 2)*x^2)*P[n - 2, x] + (3*n - 3 + n*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = -(9*n - 18 + (6*n - 12)*x + (n - 2)*x^2)*v[i - 2] + (3*n - 3 + n*x)*v[i - 1]); 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

Same fingerprint ?

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023