Real-rooted atlas

A111049 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(16 + 20\,x + 4\,x^{2} - 8\,n - 10\,n x - 2\,n x^{2}\right)\,P_{n-2}(x) + \left(20\,x + 8\,x^{2} - 10\,n x - 4\,n x^{2}\right)\,P_{n-2}'(x) + \left(4\,x^{2} - 2\,n x^{2}\right)\,P_{n-2}''(x) + \left(-12 - 6\,x + 6\,n + 3\,n x\right)\,P_{n-1}(x) + \left(-6\,x + 3\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 6 + 9\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
269217
32061334118
4723873771058949
5272240137351790305168519
6105614739343972524572208333283522
741608988130330332033413688526075217764882879

All 15 rows

Technical details
Node id
n-34941a8bb75ac185 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-fa36d41190ef9b42

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,       1;
  2:     6,       9,        2;
  3:    20,      61,       33,        4;
  4:    72,     387,      377,      105,        8;
  5:   272,    2401,     3735,     1790,      305,       16;
  6:  1056,   14739,    34397,    25245,     7220,      833,      32;
  7:  4160,   89881,   303303,   320334,   136885,    26075,    2177,     64;
  8: 16512,  545547,  2600117,  3804045,  2280908,   638386,   86947,   5505,   128;
  9: 65792, 3300001, 21865695, 43205110, 34958805, 13493802, 2667210, 272940, 13569, 256;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(8*n-16+(10*n-20)*x+(2*n-4)*x^2)*P(n-2,x) - ((10*n-20)*x+(4*n-8)*x^2)*d/dx P(n-2,x) - (2*n-4)*x^2*d^2/dx^2 P(n-2,x) + (6*n-12+(3*n-6)*x)*P(n-1,x) + (3*n-6)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 6 + 9*x + 2*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 6 + 9*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n - 16 + (10*n - 20)*x + (2*n - 4)*x^2)*P[n - 2, x] - ((10*n - 20)*x + (4*n - 8)*x^2)*D[P[n - 2, x], x] - (2*n - 4)*x^2*D[P[n - 2, x], {x, 2}] + (6*n - 12 + (3*n - 6)*x)*P[n - 1, x] + (3*n - 6)*x*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 6 + 9*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(8*n - 16 + (10*n - 20)*x + (2*n - 4)*x^2)*v[i - 2] - ((10*n - 20)*x + (4*n - 8)*x^2)*deriv(v[i - 2], x) - (2*n - 4)*x^2*deriv(deriv(v[i - 2], x), x) + (6*n - 12 + (3*n - 6)*x)*v[i - 1] + (3*n - 6)*x*deriv(v[i - 1], x))/(n - 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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