Real-rooted atlas

A086645 → basis A007318 → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-14 - 6\,x + 7\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-4 - 6\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2314
31710128
49987211208
55776762623611552
633634925263061055111584
71960134446234397684121578186464
8114243234227193683821871875321771051645376

All 15 rows

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(P_{n}(x) = \left(-1 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(6 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)
Technical details
Node id
n-f455dcc658a36426 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-aed905df62c8265e)
Fingerprint
f6-948c8082b8d2c9f1

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,        1;
   3:      17,       10,        1;
   4:      99,       87,       21,       1;
   5:     577,      676,      262,      36,       1;
   6:    3363,     4925,     2630,     610,      55,      1;
   7:   19601,    34446,    23439,    7684,    1215,     78,     1;
   8:  114243,   234227,   193683,   82187,   18753,   2177,   105,    1;
   9:  665857,  1560200,  1518620,  791480,  236870,  40376,  3612,  136,   1;
  10: 3880899, 10230201, 11455740, 7082340, 2627370, 594126, 79212, 5652, 171, 1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (7*n-14+(3*n-6)*x)*P(n-1,x) - (4+6*x+2*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[((7*n - 14 + (3*n - 6)*x)*P[n - 1, x] - (4 + 6*x + 2*x^2)*D[P[n - 1, x], x])/(n - 2)]];
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 + x; for(n = 3, N, my(i = n); v[i] = ((7*n - 14 + (3*n - 6)*x)*v[i - 1] - (4 + 6*x + 2*x^2)*deriv(v[i - 1], x))/(n - 2)); 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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023