Real-rooted atlas

A008517 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
21101526
3125105105236
415649012609452752
5111919189450173251039539208
61246682556980190575270270135135660032
7150122935302995163663540990954729725202702512818912

All 15 rows

Technical details
Node id
n-affbcb84bac5cb29 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-014ae34d4ff839c2

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,    3;
  2: 1,   10,     15;
  3: 1,   25,    105,     105;
  4: 1,   56,    490,    1260,      945;
  5: 1,  119,   1918,    9450,    17325,     10395;
  6: 1,  246,   6825,   56980,   190575,    270270,     135135;
  7: 1,  501,  22935,  302995,  1636635,   4099095,    4729725,    2027025;
  8: 1, 1012,  74316, 1487200, 12122110,  47507460,   94594500,   91891800,   34459425;
  9: 1, 2035, 235092, 6914908, 81431350, 466876410, 1422280860, 2343240900, 1964187225, 654729075;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(n+4)*x+(2*n+5)*x^2+(n+2)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (n + 4)*x + (2*n + 5)*x^2 + (n + 2)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], 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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (n + 4)*x + (2*n + 5)*x^2 + (n + 2)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + x^4)*deriv(v[i - 1], x))/(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