Real-rooted atlas

A091042 → basis A007318 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-21 - 9\,x + 14\,n + 6\,n x\right)\,P_{n-1}(x) + \left(-8 - 12\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
27310
34130576
4239245777568
51393183684615694240
6811913079794222222751131648
747321900906789925532492744213236224
8275807605865545355257805679659723665151763200

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(-1 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(6 + 2\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 7 + 3\,x\)
Technical details
Node id
n-fe8e8f3388df4395 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-c1114715384c2ac1)
Fingerprint
f6-5c55a539d6259f58

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:       7,        3;
   3:      41,       30,        5;
   4:     239,      245,       77,        7;
   5:    1393,     1836,      846,      156,       9;
   6:    8119,    13079,     7942,     2222,     275,      11;
   7:   47321,    90090,    67899,    25532,    4927,     442,     13;
   8:  275807,   605865,   545355,   257805,   67965,    9723,    665,    15;
   9: 1607521,  4002072,  4188460,  2386120,  799510,  158440,  17612,   952,   17;
  10: 9369319, 26069995, 31102012, 20739260, 8440370, 2146202, 334476, 29868, 1311, 19;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = (14*n-21+(6*n-9)*x)*P(n-1,x) - (8+12*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[((14*n - 21 + (6*n - 9)*x)*P[n - 1, x] - (8 + 12*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] = ((14*n - 21 + (6*n - 9)*x)*v[i - 1] - (8 + 12*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

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