Real-rooted atlas

A056241 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 4\,x - x^{2}\right)\,P_{n-2}(x) + \left(-4\,x - 2\,x^{2}\right)\,P_{n-2}'(x) + \left(-x^{2}\right)\,P_{n-2}''(x) + \left(4 + 2\,x\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
256112
3143112158
441152104201318
51227217802603011914
636533425377275654542112428
710941524735112258937742101556186160
832816881622107222492094858185921736721633348
998423083531357176185144810540922875623981627849014911164
10295251374966818400114668120109411303971380758082781204245110140009680
11885746111007487186921130030331080812925061586212722600179447414302262151321341284904
12265721271003282873002168525902201028941339608053632195042200359422803898851247544880015613039391288
137971621200020651682504772633157839695232074346984172923279489090465187137691823446789889140898012116182128189168648
14239148553084153498022718094645089682886231262075774786433263802750272310970951304194341299121602581015098083649740162892101271669964208
Technical details
Node id
n-a73bf794859ac224 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-099df3ed915090b2

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:    2,      1;
  2:    5,      6,       1;
  3:   14,     31,      12,       1;
  4:   41,    152,     104,      20,       1;
  5:  122,    721,     780,     260,      30,      1;
  6:  365,   3342,    5377,    2756,     545,     42,     1;
  7: 1094,  15247,   35112,   25893,    7742,   1015,    56,    1;
  8: 3281,  68816,  221072,  224920,   94858,  18592,  1736,   72,  1;
  9: 9842, 308353, 1357176, 1851448, 1054092, 287562, 39816, 2784, 90, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(3+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) + (4+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[-(3 + 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}] + (4 + 2*x)*P[n - 1, x] + 2*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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = -(3 + 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) + (4 + 2*x)*v[i - 1] + 2*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