Real-rooted atlas

A257618 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x) + \left(8\,x + 24\,x^{2} + 24\,x^{3} + 8\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
244848100
3849614409602904
4164992318725376026880117520
53249984633600203520024192009676806105696
664499968120046086558720013923840012773376042577920387641920
7128499993622208256019453593606570547200103774003207749181440221405184029083622784

All 15 rows

Technical details
Node id
n-2e3a2796ffe64916 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d6ac9abcd3bcb1be

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:   2,         4;
  2:   4,        48,          48;
  3:   8,       496,        1440,           960;
  4:  16,      4992,       31872,         53760,          26880;
  5:  32,     49984,      633600,       2035200,        2419200,         967680;
  6:  64,    499968,    12004608,      65587200,      139238400,      127733760,       42577920;
  7: 128,   4999936,   222082560,    1945359360,     6570547200,    10377400320,     7749181440,     2214051840;
  8: 256,  49999872,  4057485312,   55020994560,   277868666880,   672390512640,   844064686080,   531372441600,   132843110400;
  9: 512, 499999744, 73634734080, 1511695564800, 10988122521600, 38243673538560, 71788416860160, 74710965288960, 40649991782400, 9033331507200;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (2+8*x+10*x^2+4*x^3)*P(n-1,x) + (8*x+24*x^2+24*x^3+8*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[((2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x] + (8*x + 24*x^2 + 24*x^3 + 8*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] = ((2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1] + (8*x + 24*x^2 + 24*x^3 + 8*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