Real-rooted atlas

chebyshev_u → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0)coefficients of one signHadamard kernel

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 12\,x^{2} + 10\,x^{3} + 3\,x^{4} + n + 2\,n x - 2\,n x^{2} - 6\,n x^{3} - 3\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 9\,x^{2} + 15\,x^{3} + 11\,x^{4} + 3\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
23141532
34327656168
4560234372209880
56100564144017187804608
671541169423680097642291124128
7822421841048027800416643311210864126336

All 15 rows

Technical details
Node id
n-12d187bc73368c94 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0878106a64e78b92

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  2,   4;
  2:  3,  14,   15;
  3:  4,  32,   76,    56;
  4:  5,  60,  234,   372,    209;
  5:  6, 100,  564,  1440,   1718,    780;
  6:  7, 154, 1169,  4236,   8009,   7642,   2911;
  7:  8, 224, 2184, 10480,  27800,  41664,  33112,  10864;
  8:  9, 312, 3780, 22968,  79870, 166632, 206676, 140712,  40545;
  9: 10, 420, 6168, 45984, 200668, 545000, 935640, 989376, 589098, 151316;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (n+1+(2*n+6)*x-(2*n-12)*x^2-(6*n-10)*x^3-(3*n-3)*x^4)*P(n-1,x) + (2*x+9*x^2+15*x^3+11*x^4+3*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + 1 + (2*n + 6)*x - (2*n - 12)*x^2 - (6*n - 10)*x^3 - (3*n - 3)*x^4)*P[n - 1, x] + (2*x + 9*x^2 + 15*x^3 + 11*x^4 + 3*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + 1 + (2*n + 6)*x - (2*n - 12)*x^2 - (6*n - 10)*x^3 - (3*n - 3)*x^4)*v[i - 1] + (2*x + 9*x^2 + 15*x^3 + 11*x^4 + 3*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023