Real-rooted atlas

A059344 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
29615
3251843
4241432108781
5104120405403621
6106813951024300324077731
76064924834616254022680494215
865878545689285223960163296013608012220713
94540321352948324315096814152320122472098363161
1051972841646554150123018588067835880012859560073483202743015591
11415198521586024989011978069400694702008013696452008083152026651014611
1249888085291088597964963252915369002821499049609140697720011639738880484989120824821752085
1344847866545112327607152836219661849803312041166720111753617976014711336640063048585609373085694493
1456368395356121324886099062966888299056681240514489214006115053128184013026888594720119161826784037829151360318035716175451
Technical details
Node id
n-264b8b71d4b9f777 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-fd12dfaec634c7e4

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:       9,        6;
  3:      25,       18;
  4:     241,      432,      108;
  5:    1041,     2040,      540;
  6:   10681,    39510,    24300,     3240;
  7:   60649,   248346,   162540,    22680;
  8:  658785,  4568928,  5223960,  1632960,  136080;
  9: 4540321, 35294832, 43150968, 14152320, 1224720;
%F Formula
The row polynomials satisfy P(n,x) = (8*n-8+(6*n-6)*x)*P(n-2,x) + (6*n-6)*x*d/dx P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(8*n - 8 + (6*n - 6)*x)*P[n - 2, x] + (6*n - 6)*x*D[P[n - 2, x], x] + P[n - 1, 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; for(n = 2, N, my(i = n + 1); v[i] = (8*n - 8 + (6*n - 6)*x)*v[i - 2] + (6*n - 6)*x*deriv(v[i - 2], x) + v[i - 1]); 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