Real-rooted atlas

A091044 → basis A013610 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-3 - 3\,x + 13\,n + 9\,n x\right)\,P_{n-1}(x) + \left(-8 - 14\,x - 6\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
110616
29110227220
382013326841082944
473811568411934378040539184
56643017437817593284132186301458520960
65978711868346234944115069245120018505051036924736
753808401950554429384424236725201089612028168563674161749692041216

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(-9 - 18\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 10 + 6\,x\)
Technical details
Node id
n-5a8e43728bc85b19 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-45e577e73ddd2222)
Fingerprint
f6-cfab725842e1b6be

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:        10,          6;
  2:        91,        102,         27;
  3:       820,       1332,        684,        108;
  4:      7381,      15684,      11934,       3780,        405;
  5:     66430,     174378,     175932,      84132,      18630,       1458;
  6:    597871,    1868346,    2349441,    1506924,     512001,      85050,      5103;
  7:   5380840,   19505544,   29384424,   23672520,   10896120,    2816856,    367416,    17496;
  8:  48427561,  199763688,  350721468,  340364376,  198118710,   70001496,  14390460,  1522152,   59049;
  9: 435848050, 2015797230, 4041809640, 4590450360, 3234743100, 1454192676, 411838344, 69441624, 6101730, 196830;
%F Formula
The row polynomials satisfy n*P(n,x) = (13*n-3+(9*n-3)*x)*P(n-1,x) - (8+14*x+6*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((13*n - 3 + (9*n - 3)*x)*P[n - 1, x] - (8 + 14*x + 6*x^2)*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((13*n - 3 + (9*n - 3)*x)*v[i - 1] - (8 + 14*x + 6*x^2)*deriv(v[i - 1], x))/(n)); 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