Real-rooted atlas

A330510 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-1 - 5\,x - 10\,x^{2} - 9\,x^{3} - 3\,x^{4}\right)\,P_{n-2}(x) + \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 3 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1369
24181840
35378154177
4664226324162782
57100501114412154863453
68146966309851944374145815244
7920316947113166302196115309437467293

All 15 rows

Technical details
Node id
n-4324dc8d94b46a38 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-db9ee1b7b985cf8f

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:  2;
  1:  3,   6;
  2:  4,  18,   18;
  3:  5,  37,   81,    54;
  4:  6,  64,  226,   324,    162;
  5:  7, 100,  501,  1144,   1215,    486;
  6:  8, 146,  966,  3098,   5194,   4374,   1458;
  7:  9, 203, 1694,  7113,  16630,  21961,  15309,   4374;
  8: 10, 272, 2772, 14568,  44326,  81192,  88300,  52488,  13122;
  9: 11, 354, 4302, 27420, 103873, 246498, 370286, 341992, 177147, 39366;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(1+5*x+10*x^2+9*x^3+3*x^4)*P(n-2,x) + (2+8*x+10*x^2+4*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 6*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 3 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + 5*x + 10*x^2 + 9*x^3 + 3*x^4)*P[n - 2, x] + (2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(1 + 5*x + 10*x^2 + 9*x^3 + 3*x^4)*v[i - 2] + (2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1])/(1 + 2*x + 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. [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