Real-rooted atlas

A119900 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2410317
384028480
416148186605415
532528107659611062348
6641848578849361526182714351
712864002982436928170883360280894016
8256220321495602592641689404884066364089655945
9512755847367521744752154092061535212141612072570104847940
1010242587843585936114009601329623270522001896796271392205927701137784697
11204888499217309824729464321102510727560435226634928514465655783233352101212309370512
12409630244488307715245960518488768532877182508834650963286779056126085961071356517661300132652243015
13819210331904397110848286289465669901609287593684000425861475213448900802513995122845700019453727753216381423739576428
141638435286912189276409617680629376541137734087261331875250104983600195528837124572177936661783200599822083369704112658203015221291083991
Technical details
Node id
n-2b7cd3c06c1326ff · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-b5e14b9c92721dcc

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,     2;
  2:   4,    10,      3;
  3:   8,    40,     28,       4;
  4:  16,   148,    186,      60,       5;
  5:  32,   528,   1076,     596,     110,      6;
  6:  64,  1848,   5788,    4936,    1526,    182,      7;
  7: 128,  6400,  29824,   36928,   17088,   3360,    280,     8;
  8: 256, 22032, 149560,  259264,  168940,  48840,   6636,   408,   9;
  9: 512, 75584, 736752, 1744752, 1540920, 615352, 121416, 12072, 570, 10;
%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) + (2+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 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 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}] + (2 + 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 + 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) + (2 + 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