Real-rooted atlas

A081577 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 6\,x\right)\,P_{n-2}(x) + \left(6\,x\right)\,P_{n-2}'(x) + \left(5 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1538
23345987
3205531252271015
412895691481511888113064
580855805978264332915103243183045
6507375761351165527760644196344206557292770771
7318365562262716462836154807475982822104198480190218744991758

All 15 rows

Technical details
Node id
n-27d3d6089fc3c7ab · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5c0949063c0d56c9

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,         3;
  2:       33,        45,          9;
  3:      205,       531,        252,         27;
  4:     1289,      5691,       4815,       1188,         81;
  5:     8085,     58059,      78264,      33291,       5103,        243;
  6:    50737,    576135,    1165527,     760644,     196344,      20655,       729;
  7:   318365,   5622627,   16462836,   15480747,    5982822,    1041984,     80190,     2187;
  8:  1997721,  54306423,  224726427,  292888872,  158997087,   40751100,   5126328,   301806,    6561;
  9: 12535525, 521071515, 2997902448, 5275899927, 3865951881, 1367505585, 249939108, 23816430, 1108809, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (8+6*x)*P(n-2,x) + 6*x*d/dx P(n-2,x) + (5+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 3*x;
P[n_, x_] := P[n, x] = Expand[(8 + 6*x)*P[n - 2, x] + 6*x*D[P[n - 2, x], x] + (5 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];
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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (8 + 6*x)*v[i - 2] + 6*x*deriv(v[i - 2], x) + (5 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 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

  • 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