Real-rooted atlas

A085478 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-3 - 6\,x - 3\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(-3 - 22\,x - 45\,x^{2} - 36\,x^{3} - 10\,x^{4} + 2\,n + 13\,n x + 25\,n x^{2} + 19\,n x^{3} + 5\,n x^{4}\right)\,P_{n-1}(x) + \left(-x - 8\,x^{2} - 18\,x^{3} - 16\,x^{4} - 5\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
315511
418191341
5111426534153
61147418321089571
71171153947176542332131
81201657251825262219856107953

All 15 rows

Technical details
Node id
n-a99b6ab79c8d82f3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-525575d96ecebcc3

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: 1,  2;
   3: 1,  5,   5;
   4: 1,  8,  19,   13;
   5: 1, 11,  42,   65,   34;
   6: 1, 14,  74,  183,  210,    89;
   7: 1, 17, 115,  394,  717,   654,   233;
   8: 1, 20, 165,  725, 1825,  2622,  1985,   610;
   9: 1, 23, 224, 1203, 3885,  7703,  9134,  5911,  1597;
  10: 1, 26, 292, 1855, 7329, 18633, 30418, 30691, 17345, 4181;
%F Formula
The row polynomials satisfy (2*n-3+(4*n-6)*x+(2*n-3)*x^2)*P(n,x) = (2*n-3+(13*n-22)*x+(25*n-45)*x^2+(19*n-36)*x^3+(5*n-10)*x^4)*P(n-1,x) - (x+8*x^2+18*x^3+16*x^4+5*x^5)*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[((2*n - 3 + (13*n - 22)*x + (25*n - 45)*x^2 + (19*n - 36)*x^3 + (5*n - 10)*x^4)*P[n - 1, x] - (x + 8*x^2 + 18*x^3 + 16*x^4 + 5*x^5)*D[P[n - 1, x], x])/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*x^2)]];
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] = ((2*n - 3 + (13*n - 22)*x + (25*n - 45)*x^2 + (19*n - 36)*x^3 + (5*n - 10)*x^4)*v[i - 1] - (x + 8*x^2 + 18*x^3 + 16*x^4 + 5*x^5)*deriv(v[i - 1], x))/(2*n - 3 + (4*n - 6)*x + (2*n - 3)*x^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

Same fingerprint ?

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