Real-rooted atlas

A116414 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-6 + 6\,n\right)\,P_{n}(x) = \left(30 + 18\,x + 15\,n + 9\,n x\right)\,P_{n-1}(x) + \left(13 + 30\,x + 9\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2301848
320727081558
41230253816203245712
5675619260192788100121554609
635370129222178200116640364504374500256
717930180055014146651271700616005153090153094450620

All 15 rows

Technical details
Node id
n-f896db6f0dd1ac21 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-12773e8e19c3f694

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       0;
  1:       3;
  2:      30,       18;
  3:     207,      270,       81;
  4:    1230,     2538,     1620,      324;
  5:    6756,    19260,    19278,     8100,     1215;
  6:   35370,   129222,   178200,   116640,    36450,     4374;
  7:  179301,   800550,  1414665,  1271700,   616005,   153090,    15309;
  8:  888150,  4690890, 10141200, 11667240,  7739550,  2969946,   612360,   52488;
  9: 4323465, 26381700, 67571010, 95085900, 80601885, 42252840, 13410684, 2361960, 177147;
%F Formula
The row polynomials satisfy (6*n-6)*P(n,x) = (15*n+30+(9*n+18)*x)*P(n-1,x) + (13+30*x+9*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[((15*n + 30 + (9*n + 18)*x)*P[n - 1, x] + (13 + 30*x + 9*x^2)*D[P[n - 1, x], x])/(6*n - 6)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((15*n + 30 + (9*n + 18)*x)*v[i - 1] + (13 + 30*x + 9*x^2)*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 - 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