Real-rooted atlas

A199881 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-2}(x) + \left(x + x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1 + 2\,x\)
\(P_{1}(x) = x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0123
10123
201449
30027615
40017151033
5000316281663
6000111385326129
70000430829742255

All 15 rows

Technical details
Node id
n-22333aafcaf54c94 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-92bdcb9f4d02545f

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1, 2;
  1: 0, 1, 2;
  2: 0, 1, 4, 4;
  3: 0, 0, 2, 7,  6;
  4: 0, 0, 1, 7, 15, 10;
  5: 0, 0, 0, 3, 16, 28, 16;
  6: 0, 0, 0, 1, 11, 38, 53,  26;
  7: 0, 0, 0, 0,  4, 30, 82,  97,  42;
  8: 0, 0, 0, 0,  1, 16, 79, 173, 176,  68;
  9: 0, 0, 0, 0,  0,  5, 50, 191, 352, 315, 110;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (x+2*x^2+x^3)*P(n-2,x) + (x+x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1 + 2*x, P(1,x) = x + 2*x^2.
%t Mathematica
P[0, x_] := 1 + 2*x;
P[1, x_] := x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((x + 2*x^2 + x^3)*P[n - 2, x] + (x + x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 2*x; v[2] = x + 2*x^2; for(n = 2, N, my(i = n + 1); v[i] = ((x + 2*x^2 + x^3)*v[i - 2] + (x + x^2)*v[i - 1])/(1 + x)); 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

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