Real-rooted atlas

A144442 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5\,x + 15\,x^{2} + 15\,x^{3} + 5\,x^{4}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(5\,x + 15\,x^{2} + 15\,x^{3} + 5\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
21191939
31121357238717
4173348748282414118031
5144055935519339023073092292580173
6126437687410383453581987407537767251258922797479
71158629776859969920280238624360378474822283025078808643081058836077

All 15 rows

Technical details
Node id
n-e0355357b9c46242 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-83f6dcff4e22986d

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,      19,        19;
  3: 1,     121,       357,         238;
  4: 1,     733,      4874,        8282,         4141;
  5: 1,    4405,     59355,      193390,       230730,        92292;
  6: 1,   26437,    687410,     3834535,      8198740,      7537767,       2512589;
  7: 1,  158629,   7768599,    69920280,    238624360,    378474822,     283025078,      80864308;
  8: 1,  951781,  86697182,  1215516903,   6222366045,  15150247667,   19071280147,   12018169364,   3004542341;
  9: 1, 5710693, 961124619, 20528272772, 151721918691, 532341215532, 1003351867476, 1046242561308, 569677949379, 126595099862;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5*x+15*x^2+15*x^3+5*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (5*x+15*x^2+15*x^3+5*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[((5*x + 15*x^2 + 15*x^3 + 5*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (5*x + 15*x^2 + 15*x^3 + 5*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] = ((5*x + 15*x^2 + 15*x^3 + 5*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (5*x + 15*x^2 + 15*x^3 + 5*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