Real-rooted atlas

A116414 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(138\,n - 18\,n x - 54\,n x^{2} + 54\,n^{3}\right)\,P_{n}(x) = \left(363 - 27\,x - 243\,x^{2} - 81\,x^{3} + 633\,n + 135\,n x - 243\,n x^{2} - 81\,n x^{3} + 405\,n^{2} + 243\,n^{2} x + 135\,n^{3} + 81\,n^{3} x\right)\,P_{n-1}(x) + \left(26 - 213\,x - 729\,x^{2} - 459\,x^{3} - 81\,x^{4} + 117\,n + 270\,n x + 81\,n x^{2} + 117\,n^{2} + 270\,n^{2} x + 81\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
18311
252489109
3302477216271022
4163937682889864819241
58516260162890814526324024381449
6429431645922428111760406561011664729704389
7211886979851181494017432559339302762914082421876003164

All 15 rows

Technical details
Node id
n-b185bfd2a6292f76 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8d7c5c07c3e18258

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:       8,        3;
  2:      52,       48,        9;
  3:     302,      477,      216,        27;
  4:    1639,     3768,     2889,       864,        81;
  5:    8516,    26016,    28908,     14526,      3240,      243;
  6:   42943,   164592,   242811,    176040,     65610,    11664,      729;
  7:  211886,   979851,  1814940,   1743255,    933930,   276291,    40824,    2187;
  8: 1028356,  5579040, 12486645,  15047640,  10656360,  4517856,  1107351,  139968,   6561;
  9: 4926740, 30705165, 80761860, 117609570, 104373360, 58373217, 20452824, 4277772, 472392, 19683;
%F Formula
The row polynomials satisfy (54*n^3+138*n-18*n*x-54*n*x^2)*P(n,x) = (135*n^3+405*n^2+633*n+363+(81*n^3+243*n^2+135*n-27)*x-(243*n+243)*x^2-(81*n+81)*x^3)*P(n-1,x) + (117*n^2+117*n+26+(270*n^2+270*n-213)*x+(81*n^2+81*n-729)*x^2-459*x^3-81*x^4)*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[((135*n^3 + 405*n^2 + 633*n + 363 + (81*n^3 + 243*n^2 + 135*n - 27)*x - (243*n + 243)*x^2 - (81*n + 81)*x^3)*P[n - 1, x] + (117*n^2 + 117*n + 26 + (270*n^2 + 270*n - 213)*x + (81*n^2 + 81*n - 729)*x^2 - 459*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 + 138*n - 18*n*x - 54*n*x^2)]];
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] = ((135*n^3 + 405*n^2 + 633*n + 363 + (81*n^3 + 243*n^2 + 135*n - 27)*x - (243*n + 243)*x^2 - (81*n + 81)*x^3)*v[i - 1] + (117*n^2 + 117*n + 26 + (270*n^2 + 270*n - 213)*x + (81*n^2 + 81*n - 729)*x^2 - 459*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 + 138*n - 18*n*x - 54*n*x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023