Real-rooted atlas

A079510 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1022
203912
300204464
40015160265440
5000210134418543408
60001052695123831483330016
70000252033992124648133496294656

All 15 rows

Technical details
Node id
n-064d20684eb59b67 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3f24d40b8bba79ee

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: 0, 2;
  2: 0, 3,  9;
  3: 0, 0, 20,  44;
  4: 0, 0, 15, 160,  265;
  5: 0, 0,  0, 210, 1344,  1854;
  6: 0, 0,  0, 105, 2695, 12383,  14833;
  7: 0, 0,  0,   0, 2520, 33992, 124648,  133496;
  8: 0, 0,  0,   0,  945, 47880, 441630, 1366776,  1334961;
  9: 0, 0,  0,   0,    0, 34650, 843920, 6002700, 16249200, 14684570;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = ((n+1)*x+(2*n+2)*x^2+(n+1)*x^3)*P(n-2,x) + ((n+1)*x+(n+1)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(((n + 1)*x + (2*n + 2)*x^2 + (n + 1)*x^3)*P[n - 2, x] + ((n + 1)*x + (n + 1)*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; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (((n + 1)*x + (2*n + 2)*x^2 + (n + 1)*x^3)*v[i - 2] + ((n + 1)*x + (n + 1)*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

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