Real-rooted atlas

A091044 → basis A007318 → basis A013610

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 6\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16612
2356627128
32045644681081344
4118943325454270040514080
56930313385302840932137701458147456
6403912181424640494893482625216463851031544192
7235416147828037897205079240377460015104882857681749616171008

All 15 rows

Technical details
Node id
n-2229d617a5e5f9be · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-c99502c0ad94acbf

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 0)
  • 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:       6,        6;
  2:      35,       66,        27;
  3:     204,      564,       468,       108;
  4:    1189,     4332,      5454,      2700,       405;
  5:    6930,    31338,     53028,     40932,     13770,      1458;
  6:   40391,   218142,    464049,    489348,    262521,     64638,      5103;
  7:  235416,  1478280,   3789720,   5079240,   3774600,   1510488,    285768,    17496;
  8: 1372105,  9821688,  29471580,  47976840,  45747990,  25666632,   8021916,  1207224,   59049;
  9: 7997214, 64271982, 220981464, 423568440, 493991460, 358616484, 158811192, 40048344, 4920750, 196830;
%F Formula
The row polynomials satisfy P(n,x) = -(1+6*x+9*x^2)*P(n-2,x) + (6+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 6*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 6*x;
P[n_, x_] := P[n, x] = Expand[-(1 + 6*x + 9*x^2)*P[n - 2, x] + (6 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(1 + 6*x + 9*x^2)*v[i - 2] + (6 + 6*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 - 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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023