Real-rooted atlas

A122542 → x ↦ x+1 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2213
387116
4284113183
5100221117201459
635611358992552812674
7126856596319264647737116407
845162766541985243696408808471105799

All 15 rows

Technical details
Node id
n-279334a58393797d · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-4139c2f8a6e84b16

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:     2,      1;
   3:     8,      7,       1;
   4:    28,     41,      13,       1;
   5:   100,    221,     117,      20,      1;
   6:   356,   1135,     899,     255,     28,      1;
   7:  1268,   5659,    6319,    2646,    477,     37,     1;
   8:  4516,  27665,   41985,   24369,   6408,    808,    47,    1;
   9: 16084, 133421,  268525,  207748,  74733,  13608,  1276,   58,  1;
  10: 57284, 637279, 1671651, 1678843, 793696, 195661, 26276, 1912, 70, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-2,x) + x*d/dx P(n-2,x) + (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (3 + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = (2 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (3 + x)*v[i - 1] + 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, v[i])))};
T(10)

polytool c48629374597

Properties

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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