Real-rooted atlas

A078812 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-6 + 6\,n\right)\,P_{n}(x) = \left(-18 - 36\,x - 10\,x^{2} + 18\,n + 27\,n x + 5\,n x^{2}\right)\,P_{n-1}(x) + \left(-9\,x - 18\,x^{2} - 5\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2336
3918835
427817521204
581324459270551189
62431215232221069001446930
7729437410530129608559286237740391
8218715309444696925562235321578817987235416
96561524881786053367983839402702161143632655025841372105
101968317714769109215309002117745189321310915023905827858567657997214
11590495904902598156661348810767330116960768570205417603612931382295001771146611179
1217714719486179546255274577855145136265859318586810983634551015316020417733266304846368271669860
13531441637729234425567110421630234129285345318552362853918273383748146414385543040201322800218987481213931583407981
14159432320726199122231430432592974102410649017108857262073066048184285368012007133015658097051872800464120291853973753178119228778026
1547829696696156642834144616580959204335672825809455501811117726037114024406328767847025502618527021128548686311448721265728771524681083204053789260175
Technical details
Node id
n-a1318ff36656ed88 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ba43151883801832

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,      3;
   3:     9,     18,      8;
   4:    27,     81,     75,      21;
   5:    81,    324,    459,     270,      55;
   6:   243,   1215,   2322,    2106,     900,     144;
   7:   729,   4374,  10530,   12960,    8559,    2862,     377;
   8:  2187,  15309,  44469,   69255,   62235,   32157,    8817,    987;
   9:  6561,  52488, 178605,  336798,  383940,  270216,  114363,  26550,  2584;
  10: 19683, 177147, 691092, 1530900, 2117745, 1893213, 1091502, 390582, 78585, 6765;
%F Formula
The row polynomials satisfy (6*n-6)*P(n,x) = (18*n-18+(27*n-36)*x+(5*n-10)*x^2)*P(n-1,x) - (9*x+18*x^2+5*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((18*n - 18 + (27*n - 36)*x + (5*n - 10)*x^2)*P[n - 1, x] - (9*x + 18*x^2 + 5*x^3)*D[P[n - 1, x], x])/(6*n - 6)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((18*n - 18 + (27*n - 36)*x + (5*n - 10)*x^2)*v[i - 1] - (9*x + 18*x^2 + 5*x^3)*deriv(v[i - 1], x))/(6*n - 6)); 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