Real-rooted atlas

A123097 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 10\,x + 2\,x^{2} - 4\,n - 5\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(10\,x + 4\,x^{2} - 5\,n x - 2\,n x^{2}\right)\,P_{n-2}'(x) + \left(2\,x^{2} - n x^{2}\right)\,P_{n-2}''(x) + \left(-8 - 4\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-4\,x + 2\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 2\,x\)
\(P_{2}(x) = 14 + 16\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
21416333
33681374158
488344284715792
52081331176576412164195
64804866974264951735190723515
710881713749903481181928534992818139319

All 15 rows

Technical details
Node id
n-ba874729867b2207 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-2726951d0c2185c2

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:    5,      2;
  2:   14,     16,       3;
  3:   36,     81,      37,       4;
  4:   88,    344,     284,      71,       5;
  5:  208,   1331,    1765,     764,     121,      6;
  6:  480,   4866,    9742,    6495,    1735,    190,      7;
  7: 1088,  17137,   49903,   48118,   19285,   3499,    281,     8;
  8: 2432,  58804,  243084,  326067,  184009,  49196,   6461,   397,   9;
  9: 5376, 198015, 1142785, 2077624, 1586781, 586488, 112014, 11144, 541, 10;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(5*n-10)*x+(n-2)*x^2)*P(n-2,x) - ((5*n-10)*x+(2*n-4)*x^2)*d/dx P(n-2,x) - (n-2)*x^2*d^2/dx^2 P(n-2,x) + (4*n-8+(2*n-4)*x)*P(n-1,x) + (2*n-4)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 2*x, P(2,x) = 14 + 16*x + 3*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[2, x_] := 14 + 16*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (5*n - 10)*x + (n - 2)*x^2)*P[n - 2, x] - ((5*n - 10)*x + (2*n - 4)*x^2)*D[P[n - 2, x], x] - (n - 2)*x^2*D[P[n - 2, x], {x, 2}] + (4*n - 8 + (2*n - 4)*x)*P[n - 1, x] + (2*n - 4)*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] = 5 + 2*x; v[3] = 14 + 16*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (5*n - 10)*x + (n - 2)*x^2)*v[i - 2] - ((5*n - 10)*x + (2*n - 4)*x^2)*deriv(v[i - 2], x) - (n - 2)*x^2*deriv(deriv(v[i - 2], x), x) + (4*n - 8 + (2*n - 4)*x)*v[i - 1] + (2*n - 4)*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