Real-rooted atlas

A122848 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(3 + 11\,x + 11\,x^{2} + 3\,x^{3} - 2\,n - 7\,n x - 8\,n x^{2} - 3\,n x^{3}\right)\,P_{n-1}(x) + \left(1 + 6\,x + 12\,x^{2} + 10\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
22101224
34307254160
410963365042701216
52631014403240351014589984
67610686120182402970024948842488576
72323780258729618020916026535618144051516833536

All 15 rows

Technical details
Node id
n-45846a9b4897e9b9 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-fdf21aecf4a8788b

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,     3;
  2:    2,    10,     12;
  3:    4,    30,     72,      54;
  4:   10,    96,    336,     504,     270;
  5:   26,   310,   1440,    3240,    3510,     1458;
  6:   76,  1068,   6120,   18240,   29700,    24948,     8424;
  7:  232,  3780,  25872,   96180,  209160,   265356,   181440,    51516;
  8:  764, 14080, 111552,  495264, 1344840,  2282112,  2358720,  1355616,   331452;
  9: 2620, 54036, 487296, 2517984, 8202600, 17440920, 24167808, 21018528, 10401372, 2230740;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(2*n-3+(7*n-11)*x+(8*n-11)*x^2+(3*n-3)*x^3)*P(n-1,x) + (1+6*x+12*x^2+10*x^3+3*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*n - 3 + (7*n - 11)*x + (8*n - 11)*x^2 + (3*n - 3)*x^3)*P[n - 1, x] + (1 + 6*x + 12*x^2 + 10*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + x)]];
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*n - 3 + (7*n - 11)*x + (8*n - 11)*x^2 + (3*n - 3)*x^3)*v[i - 1] + (1 + 6*x + 12*x^2 + 10*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + x)); 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

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