Real-rooted atlas

chebyshev_t → 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

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
216714
3112332672
41209416497376
51302106007653621968
6142405166834113426135110304
71567073906113191800414917504253952

All 15 rows

Technical details
Node id
n-e9f3812507785414 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-73a02a3902b59fb9

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: 1,  2;
  2: 1,  6,    7;
  3: 1, 12,   33,    26;
  4: 1, 20,   94,   164,    97;
  5: 1, 30,  210,   600,   765,    362;
  6: 1, 42,  405,  1668,  3411,   3426,   1351;
  7: 1, 56,  707,  3906, 11319,  18004,  14917,   5042;
  8: 1, 72, 1148,  8120, 31110,  69368,  90236,  63624,  18817;
  9: 1, 90, 1764, 15456, 74862, 218628, 395700, 435312, 267129, 70226;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (n-1+(2*n-2)*x-(2*n-2)*x^2-(6*n-6)*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 >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (2*n - 2)*x - (2*n - 2)*x^2 - (6*n - 6)*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 - 1 + (2*n - 2)*x + (n - 1)*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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (2*n - 2)*x - (2*n - 2)*x^2 - (6*n - 6)*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 - 1 + (2*n - 2)*x + (n - 1)*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