Real-rooted atlas

A113953 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n}(x) = \left(29 + 152\,x + 282\,x^{2} + 224\,x^{3} + 65\,x^{4} - 25\,n - 124\,n x - 238\,n x^{2} - 204\,n x^{3} - 65\,n x^{4}\right)\,P_{n-1}(x) + \left(9 + 85\,x + 266\,x^{2} + 378\,x^{3} + 253\,x^{4} + 65\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1156
23223560
3559215225504
411164874194014754464
5213972882993016025962538880
64396286934034810064512685062875340416
7852223242411422274824159409259728754106252975616

All 15 rows

Technical details
Node id
n-30a350309df1c375 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d9edfa0355394138

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,     5;
  2:   3,    22,     35;
  3:   5,    59,    215,     225;
  4:  11,   164,    874,    1940,    1475;
  5:  21,   397,   2882,    9930,   16025,     9625;
  6:  43,   962,   8693,   40348,  100645,   126850,    62875;
  7:  85,  2223,  24241,  142227,  482415,   940925,   972875,   410625;
  8: 171,  5088,  64716,  458064, 1965946,  5217920,  8331900,  7298000,  2681875;
  9: 341, 11409, 166164, 1379220, 7170230, 24140750, 52482500, 70862500, 53828125, 17515625;
%F Formula
The row polynomials satisfy (4*n+8*n*x+4*n*x^2)*P(n,x) = -(25*n-29+(124*n-152)*x+(238*n-282)*x^2+(204*n-224)*x^3+(65*n-65)*x^4)*P(n-1,x) + (9+85*x+266*x^2+378*x^3+253*x^4+65*x^5)*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[(-(25*n - 29 + (124*n - 152)*x + (238*n - 282)*x^2 + (204*n - 224)*x^3 + (65*n - 65)*x^4)*P[n - 1, x] + (9 + 85*x + 266*x^2 + 378*x^3 + 253*x^4 + 65*x^5)*D[P[n - 1, x], x])/(4*n + 8*n*x + 4*n*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] = (-(25*n - 29 + (124*n - 152)*x + (238*n - 282)*x^2 + (204*n - 224)*x^3 + (65*n - 65)*x^4)*v[i - 1] + (9 + 85*x + 266*x^2 + 378*x^3 + 253*x^4 + 65*x^5)*deriv(v[i - 1], x))/(4*n + 8*n*x + 4*n*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

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
  2. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741