Real-rooted atlas

A091042 → basis A013609 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-30 - 18\,x + 20\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-12 - 20\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
210616
3768020176
4568840392561856
5424080645472153614419456
6316487356865472274565280352203776
7236224648960714688399360117312166408322134016
81763200558912073363205125760203328044889649280192022347776

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(-4 - 8\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(8 + 4\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 10 + 6\,x\)
Technical details
Node id
n-a82652fb4aa6d6f1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-1ce312e71402c016)
Fingerprint
f6-77e5786089972c16

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:       10,         6;
   3:       76,        80,        20;
   4:      568,       840,       392,        56;
   5:     4240,      8064,      5472,      1536,       144;
   6:    31648,     73568,     65472,     27456,      5280,       352;
   7:   236224,    648960,    714688,    399360,    117312,     16640,      832;
   8:  1763200,   5589120,   7336320,   5125760,   2033280,    448896,    49280,    1920;
   9: 13160704,  47280128,  72051712,  60440576,  30246400,   9121792,  1584128,  139264,   4352;
  10: 98232832, 394421760, 684423168, 670181376, 405249024, 155395072, 37238784, 5253120, 379392, 9728;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = (20*n-30+(12*n-18)*x)*P(n-1,x) - (12+20*x+8*x^2)*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[((20*n - 30 + (12*n - 18)*x)*P[n - 1, x] - (12 + 20*x + 8*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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] = ((20*n - 30 + (12*n - 18)*x)*v[i - 1] - (12 + 20*x + 8*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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