Real-rooted atlas

A131436 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 6\,x - 2\,x^{2}\right)\,P_{n-2}(x) + \left(-6\,x - 4\,x^{2}\right)\,P_{n-2}'(x) + \left(-2\,x^{2}\right)\,P_{n-2}''(x) + \left(3 + 3\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
2721735
3151059015225
431465775310311612
5631953567040959456312789
612780013822744450177802667127111379
7255323852463304337552677506783071402551055700

All 15 rows

Technical details
Node id
n-b847f956136e2433 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-4732d0911d5a78c5

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:    3,      3;
  2:    7,     21,       7;
  3:   15,    105,      90,       15;
  4:   31,    465,     775,      310,       31;
  5:   63,   1953,    5670,     4095,      945,       63;
  6:  127,   8001,   38227,    44450,    17780,     2667,     127;
  7:  255,  32385,  246330,   433755,   267750,    67830,    7140,    255;
  8:  511, 130305, 1545775,  3970470,  3551961,  1352106,  236082,  18396,   511;
  9: 1023, 522753, 9544590, 34889415, 43503075, 23352021, 6015240, 767250, 46035, 1023;
%F Formula
The row polynomials satisfy P(n,x) = -(2+6*x+2*x^2)*P(n-2,x) - (6*x+4*x^2)*d/dx P(n-2,x) - 2*x^2*d^2/dx^2 P(n-2,x) + (3+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(2 + 6*x + 2*x^2)*P[n - 2, x] - (6*x + 4*x^2)*D[P[n - 2, x], x] - 2*x^2*D[P[n - 2, x], {x, 2}] + (3 + 3*x)*P[n - 1, x] + 3*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] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 6*x + 2*x^2)*v[i - 2] - (6*x + 4*x^2)*deriv(v[i - 2], x) - 2*x^2*deriv(deriv(v[i - 2], x), x) + (3 + 3*x)*v[i - 1] + 3*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

Same fingerprint ?

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