Real-rooted atlas

A086645 → basis A013610 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-26 - 18\,x + 13\,n + 9\,n x\right)\,P_{n-1}(x) + \left(-8 - 14\,x - 6\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2538
34142992
4365549243271184
532816564451811888115632
6295257381569210294305265243207488
7265721797166946647558468166455218707292757056
823914858370201120321819089577381534385560386751218736643328

All 15 rows

Other recurrences

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

\(P_{n}(x) = \left(-9 - 18\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(10 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 5 + 3\,x\)
Technical details
Node id
n-287f303befef026a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-336aec15eebed0a0)
Fingerprint
f6-f792c171e0306e3f

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:         5,         3;
   3:        41,        42,          9;
   4:       365,       549,        243,         27;
   5:      3281,      6564,       4518,       1188,         81;
   6:     29525,     73815,      69210,      29430,       5265,       243;
   7:    265721,    797166,     946647,     558468,     166455,     21870,       729;
   8:   2391485,   8370201,   12032181,    9089577,    3815343,    855603,     86751,     2187;
   9:  21523361,  86093448,  145282716,  133848504,   72620550,  23228856,   4102812,   332424,    6561;
  10: 193710245, 871696107, 1688911236, 1836464076, 1223056422, 509829066, 129881556, 18659484, 1240029, 19683;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (13*n-26+(9*n-18)*x)*P(n-1,x) - (8+14*x+6*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 3*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[((13*n - 26 + (9*n - 18)*x)*P[n - 1, x] - (8 + 14*x + 6*x^2)*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 5 + 3*x; for(n = 3, N, my(i = n); v[i] = ((13*n - 26 + (9*n - 18)*x)*v[i - 1] - (8 + 14*x + 6*x^2)*deriv(v[i - 1], x))/(n - 2)); 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