Real-rooted atlas

A046716 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(18 + 24\,x + 8\,x^{2} - 9\,n - 12\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-2 - 2\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 4 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2426
31918441
4118162688356
5102118801196288164401
61208828794258241042416643278826
718372754628264914438562411478413856641893481
833892421216517018171340144584646463824154009615315212856341416

All 15 rows

Technical details
Node id
n-284702f030aa8daf · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e0f36233ca34373d

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,          2;
   3:         19,         18,           4;
   4:        118,        162,          68,           8;
   5:       1021,       1880,        1196,         288,          16;
   6:      12088,      28794,       25824,       10424,        1664,         32;
   7:     183727,     546282,      649144,      385624,      114784,      13856,         64;
   8:    3389242,   12165170,    18171340,    14458464,     6463824,    1540096,     153152,       128;
   9:   73156249,  308508288,   556998104,   558084736,   335131184,  120611328,   24088448,   2060288,      256;
  10: 1804349548, 8753785074, 18564447552, 22477594992, 16994307648, 8215393632, 2479761920, 427285248, 32179200, 512;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n-18+(12*n-24)*x+(4*n-8)*x^2)*P(n-2,x) + (3*n-2+(2*n-2)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(9*n - 18 + (12*n - 24)*x + (4*n - 8)*x^2)*P[n - 2, x] + (3*n - 2 + (2*n - 2)*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = -(9*n - 18 + (12*n - 24)*x + (4*n - 8)*x^2)*v[i - 2] + (3*n - 2 + (2*n - 2)*x)*v[i - 1]); 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