Real-rooted atlas

A026729 → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-4 - 8\,x - 4\,x^{2} + 4\,n + 8\,n x + 4\,n x^{2}\right)\,P_{n}(x) = \left(30 + 164\,x + 328\,x^{2} + 284\,x^{3} + 90\,x^{4} - 13\,n - 68\,n x - 142\,n x^{2} - 132\,n x^{3} - 45\,n x^{4}\right)\,P_{n-1}(x) + \left(5 + 49\,x + 162\,x^{2} + 242\,x^{3} + 169\,x^{4} + 45\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2156
32163048
4337145175360
5580466116010252736
68160124847208600600020736
713310301915316425156095035125157248
8215836817434191622553547254188752056251192320

All 15 rows

Technical details
Node id
n-b8658464a7171d06 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ebeb188936acb8ca

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  1,    5;
   3:  2,   16,    30;
   4:  3,   37,   145,    175;
   5:  5,   80,   466,   1160,    1025;
   6:  8,  160,  1248,   4720,    8600,    6000;
   7: 13,  310,  3019,  15316,   42515,   60950,    35125;
   8: 21,  583,  6817,  43419,  162255,  354725,   418875,   205625;
   9: 34, 1076, 14676, 112484,  528856, 1558620,  2805900,  2815500,  1203750;
  10: 55, 1955, 30476, 273100, 1548130, 5748250, 13957500, 21337500, 18609375, 7046875;
%F Formula
The row polynomials satisfy (4*n-4+(8*n-8)*x+(4*n-4)*x^2)*P(n,x) = -(13*n-30+(68*n-164)*x+(142*n-328)*x^2+(132*n-284)*x^3+(45*n-90)*x^4)*P(n-1,x) + (5+49*x+162*x^2+242*x^3+169*x^4+45*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(13*n - 30 + (68*n - 164)*x + (142*n - 328)*x^2 + (132*n - 284)*x^3 + (45*n - 90)*x^4)*P[n - 1, x] + (5 + 49*x + 162*x^2 + 242*x^3 + 169*x^4 + 45*x^5)*D[P[n - 1, x], x])/(4*n - 4 + (8*n - 8)*x + (4*n - 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(13*n - 30 + (68*n - 164)*x + (142*n - 328)*x^2 + (132*n - 284)*x^3 + (45*n - 90)*x^4)*v[i - 1] + (5 + 49*x + 162*x^2 + 242*x^3 + 169*x^4 + 45*x^5)*deriv(v[i - 1], x))/(4*n - 4 + (8*n - 8)*x + (4*n - 4)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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