Real-rooted atlas

A111006 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2 + 4\,x + x^{2}\right)\,P_{n-2}(x) + \left(4\,x + 2\,x^{2}\right)\,P_{n-2}'(x) + \left(x^{2}\right)\,P_{n-2}''(x) + \left(1 + x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
237212
352720355
411113156555340
52141998862313082189
64315815944594121292931315944
7855803338045049127704640662621124940

All 15 rows

Technical details
Node id
n-3d2e989221bae0ab · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-3f005b735e8f3acc

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:   1,     1;
  2:   3,     7,       2;
  3:   5,    27,      20,       3;
  4:  11,   113,     156,      55,       5;
  5:  21,   419,     988,     623,     130,       8;
  6:  43,  1581,    5944,    5941,    2129,     293,     13;
  7:  85,  5803,   33804,   50491,   27704,    6406,    626,    21;
  8: 171, 21349,  188072,  403721,  318235,  109935,  17747,  1295,   34;
  9: 341, 78027, 1026116, 3089011, 3364730, 1629836, 386608, 46041, 2605, 55;
%F Formula
The row polynomials satisfy P(n,x) = (2+4*x+x^2)*P(n-2,x) + (4*x+2*x^2)*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) + (1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[(2 + 4*x + x^2)*P[n - 2, x] + (4*x + 2*x^2)*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}] + (1 + 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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 4*x + x^2)*v[i - 2] + (4*x + 2*x^2)*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), x) + (1 + 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