Real-rooted atlas

A098158 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2011
30123
400347
50018817
60005201641
7000118483299
8000075611264239

All 15 rows

Technical details
Node id
n-4d1e2be69e19d889 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-702b8764f2c773f5

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: 0, 1;
   3: 0, 1, 2;
   4: 0, 0, 3, 4;
   5: 0, 0, 1, 8,  8;
   6: 0, 0, 0, 5, 20, 16;
   7: 0, 0, 0, 1, 18, 48,  32;
   8: 0, 0, 0, 0,  7, 56, 112,  64;
   9: 0, 0, 0, 0,  1, 32, 160, 256, 128;
  10: 0, 0, 0, 0,  0,  9, 120, 432, 576, 256;
%F Formula
The row polynomials satisfy (n-2+(2*n-4)*x+(n-2)*x^2)*P(n,x) = -(n-2+(2*n-4)*x+(n-2)*x^2)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (2*n - 4)*x + (n - 2)*x^2)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n - 2 + (2*n - 4)*x + (n - 2)*x^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] = x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (2*n - 4)*x + (n - 2)*x^2)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n - 2 + (2*n - 4)*x + (n - 2)*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. [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