Real-rooted atlas

A201637 → reverse (shift +1) → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1011
20235
306201541
4024130210105469
50120924238025209456889
60720730826432441003465010395123605
705040642243036607053208662505405401351352620169

All 15 rows

Technical details
Node id
n-84b7e6675e2a3d3e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0c4bf0a352e1721b

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,      1;
  2: 0,      2,       3;
  3: 0,      6,      20,       15;
  4: 0,     24,     130,      210,       105;
  5: 0,    120,     924,     2380,      2520,       945;
  6: 0,    720,    7308,    26432,     44100,     34650,     10395;
  7: 0,   5040,   64224,   303660,    705320,    866250,    540540,    135135;
  8: 0,  40320,  623376,  3678840,  11098780,  18858840,  18288270,   9459450,   2027025;
  9: 0, 362880, 6636960, 47324376, 177331440, 389449060, 520059540, 416215800, 183783600, 34459425;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (n-1+(3*n-2)*x+(3*n-1)*x^2+n*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[((n - 1 + (3*n - 2)*x + (3*n - 1)*x^2 + n*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] = ((n - 1 + (3*n - 2)*x + (3*n - 1)*x^2 + n*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