Real-rooted atlas

A143685 → basis A007318 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(18 + 9\,x\right)\,P_{n-2}(x) + \left(9\,x\right)\,P_{n-2}'(x) + \left(3 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
22716144
3135145301311
489113904754512802
55103115816150106061123956
6313479697671941188101946781221099
71858957778657823102878514526631829612082466

All 15 rows

Technical details
Node id
n-25591ee324abb3de · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-8fad4560d38877fb

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:       3,        1;
  2:      27,       16,        1;
  3:     135,      145,       30,        1;
  4:     891,     1390,      475,       45,        1;
  5:    5103,    11581,     6150,     1060,       61,       1;
  6:   31347,    96976,    71941,    18810,     1946,      78,      1;
  7:  185895,   777865,   782310,   287851,    45266,    3182,     96,    1;
  8: 1121931,  6197830,  8152075,  4003335,   879087,   93150,   4820,  115,   1;
  9: 6711903, 48588661, 82122150, 52166170, 15191967, 2232147, 172080, 6915, 135, 1;
%F Formula
The row polynomials satisfy P(n,x) = (18+9*x)*P(n-2,x) + 9*x*d/dx P(n-2,x) + (3+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[(18 + 9*x)*P[n - 2, x] + 9*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, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (18 + 9*x)*v[i - 2] + 9*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 - 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