Real-rooted atlas

A144440 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21131327
3157165110333
412331482249812495463
5193711713375024454017816112509
6137538737847386910034829198573066192794959
71150176332255475664184054702901258621650398618582881378189

All 15 rows

Technical details
Node id
n-b0c8b82f34ff32ea · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4eb074acc1a49a5d

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,      2;
  2: 1,     13,       13;
  3: 1,     57,      165,       110;
  4: 1,    233,     1482,      2498,       1249;
  5: 1,    937,    11713,     37502,      44540,       17816;
  6: 1,   3753,    87378,    473869,    1003482,      919857,      306619;
  7: 1,  15017,   633225,   5475664,   18405470,    29012586,    21650398,     6185828;
  8: 1,  60073,  4518922,  60095833,  301187155,   726310009,   910341541,   572775604,   143193901;
  9: 1, 240297, 31977873, 639054432, 4594813845, 15909223716, 29786013600, 30959883060, 16837195095, 3741598910;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (3*x+9*x^2+9*x^3+3*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*x + 9*x^2 + 9*x^3 + 3*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*x + 9*x^2 + 9*x^3 + 3*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 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