Real-rooted atlas

A166343 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 14\,x + 14\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
11141429
21308456171
31623425602801245
41126121236084200168010827
51254401419280390403528011760109629
615101280493176291600445920329280940801267371
71102239942423920192388044251205426400338688084672016473885
812046122892185544811739000380940006896064070506240381024008467200237849867

All 15 rows

Technical details
Node id
n-5e5f6e74cb22a352 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-addc5ba4413954b4

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1,   14,      14;
   2: 1,   30,      84,       56;
   3: 1,   62,     342,      560,       280;
   4: 1,  126,    1212,     3608,      4200,       1680;
   5: 1,  254,    4014,    19280,     39040,      35280,      11760;
   6: 1,  510,   12804,    93176,    291600,     445920,     329280,       94080;
   7: 1, 1022,   39942,   423920,   1923880,    4425120,    5426400,     3386880,      846720;
   8: 1, 2046,  122892,  1855448,  11739000,   38094000,   68960640,    70506240,    38102400,     8467200;
   9: 1, 4094,  374814,  7913360,  67972240,  298998000,  749382480,  1115735040,   977477760,   465696000,   93139200;
  10: 1, 8190, 1136724, 33152696, 379428000, 2201821440, 7338663360, 14920940160, 18838915200, 14431737600, 6147187200, 1117670400;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 14*x + 14*x^2.
%t Mathematica
P[1, x_] := 1 + 14*x + 14*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 4*x + 5*x^2 + 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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 14*x + 14*x^2; for(n = 2, N, my(i = n); v[i] = ((1 + 4*x + 5*x^2 + 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, v[i])))};
T(10)

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