Real-rooted atlas

A209414 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
267114
3184115175
454215147261443
5162106511873854012840
648651118643443083557119563
71458240735917944349129781596771143711
8437411208738961140737017067332214278610011119216
91312251812124986273534217201861253811970854454212619196341
10393662383927157362432946701422211135795225114647951421047020155179404011
111180981093588197844187238755125232111910107937626264245193560931264990103951871717963850
12354294500690156027885391894078946233504620513772670644330835007696766079127734658611486122216778048941
1310628822289457053688731059147892250892282354276016776561237661359340814973241922017993121634232678006520631260166637928927
14318864610460076712246164723511408474031821818107633119717982868595705265621269310730564594549596485462406317614441253798279373011680705883046
Technical details
Node id
n-252cf4275cd87a71 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-094686ea6429288f

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:     6,      7,       1;
  3:    18,     41,      15,       1;
  4:    54,    215,     147,      26,       1;
  5:   162,   1065,    1187,     385,      40,      1;
  6:   486,   5111,    8643,    4430,     835,     57,     1;
  7:  1458,  24073,   59179,   44349,   12978,   1596,    77,    1;
  8:  4374, 112087,  389611,  407370,  170673,  32214,  2786,  100,   1;
  9: 13122, 518121, 2498627, 3534217, 2018612, 538119, 70854, 4542, 126, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(2*x+x^2)*P(n-2,x) - (2*x+2*x^2)*d/dx P(n-2,x) - x^2*d^2/dx^2 P(n-2,x) + (3+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[-(2*x + x^2)*P[n - 2, x] - (2*x + 2*x^2)*D[P[n - 2, x], x] - x^2*D[P[n - 2, x], {x, 2}] + (3 + 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] = -(2*x + x^2)*v[i - 2] - (2*x + 2*x^2)*deriv(v[i - 2], x) - x^2*deriv(deriv(v[i - 2], x), x) + (3 + 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