Real-rooted atlas

A091042 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-12 - 11\,x + 8\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2415
31612129
46480241169
5256448240401985
61024230417925606015741
7409611264115205376112084133461
8163845324867584422401344020161121195025

All 15 rows

Technical details
Node id
n-9cc539ebe6b7bba5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bfe4d29ca3d34feb

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:      4,       1;
   3:     16,      12,       1;
   4:     64,      80,      24,       1;
   5:    256,     448,     240,      40,       1;
   6:   1024,    2304,    1792,     560,      60,      1;
   7:   4096,   11264,   11520,    5376,    1120,     84,     1;
   8:  16384,   53248,   67584,   42240,   13440,   2016,   112,    1;
   9:  65536,  245760,  372736,  292864,  126720,  29568,  3360,  144,   1;
  10: 262144, 1114112, 1966080, 1863680, 1025024, 329472, 59136, 5280, 180, 1;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = (8*n-12+(6*n-11)*x)*P(n-1,x) - (4*x+4*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[((8*n - 12 + (6*n - 11)*x)*P[n - 1, x] - (4*x + 4*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] = ((8*n - 12 + (6*n - 11)*x)*v[i - 1] - (4*x + 4*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

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