Real-rooted atlas

A257615 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(5 + 20\,x + 25\,x^{2} + 10\,x^{3}\right)\,P_{n-1}(x) + \left(2\,x + 6\,x^{2} + 6\,x^{3} + 2\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
151015
225120120265
31251090252016805415
46258880357605376026880125905
5312568410428400109200012096004838403285375
6156255101204676520180096003319680029030400967680095115865
77812537270904821012026357688071971200010329984007451136002128896003026305815

All 15 rows

Technical details
Node id
n-49f8fa7d7f77a430 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-77f8f710632b75b3

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:       5,        10;
  2:      25,       120,        120;
  3:     125,      1090,       2520,        1680;
  4:     625,      8880,      35760,       53760,        26880;
  5:    3125,     68410,     428400,     1092000,      1209600,       483840;
  6:   15625,    510120,    4676520,    18009600,     33196800,     29030400,       9676800;
  7:   78125,   3727090,   48210120,   263576880,    719712000,   1032998400,     745113600,     212889600;
  8:  390625,  26870880,  478616160,  3574287360,  13573486080,  28449792000,   33326899200,   20437401600,   5109350400;
  9: 1953125, 192002410, 4629996000, 46017787200, 233643916800, 671069629440, 1135553126400, 1121502412800, 597793996800, 132843110400;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (5+20*x+25*x^2+10*x^3)*P(n-1,x) + (2*x+6*x^2+6*x^3+2*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[((5 + 20*x + 25*x^2 + 10*x^3)*P[n - 1, x] + (2*x + 6*x^2 + 6*x^3 + 2*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] = ((5 + 20*x + 25*x^2 + 10*x^3)*v[i - 1] + (2*x + 6*x^2 + 6*x^3 + 2*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