Real-rooted atlas

A153518 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(-15 - 15\,x + 5\,n + 5\,n x\right)\,P_{n-2}(x) + \left(-15\,x + 5\,n x\right)\,P_{n-2}'(x) + \left(-6 - 3\,x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-3\,x + n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 10 + 5\,x\)
\(P_{2}(x) = 50 + 52\,x + 2\,x^{2}\)
\(P_{3}(x) = 250 + 506\,x + 135\,x^{2} + 2\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
110515
250522104
32505061352893
47502538133615524781
5275014674124372826177232866
6925075902971523634747132012223567
73225039744672443539759283180706522721642197

All 15 rows

Technical details
Node id
n-5f02b0c44d7cf474 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-24712543c7b52d43

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:      2;
  1:     10,        5;
  2:     50,       52,        2;
  3:    250,      506,      135,        2;
  4:    750,     2538,     1336,      155,        2;
  5:   2750,    14674,    12437,     2826,      177,       2;
  6:   9250,    75902,    97152,    36347,     4713,     201,      2;
  7:  32250,   397446,   724435,   397592,    83180,    7065,    227,     2;
  8: 110750,  2029858,  5131976,  3925095,  1196232,  162230,   9956,   255,   2;
  9: 382750, 10336034, 35411517, 36331466, 15169947, 2959692, 285148, 13466, 285, 2;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = (5*n-15+(5*n-15)*x)*P(n-2,x) + (5*n-15)*x*d/dx P(n-2,x) + (2*n-6+(n-3)*x)*P(n-1,x) + (n-3)*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 2, P(1,x) = 10 + 5*x, P(2,x) = 50 + 52*x + 2*x^2, P(3,x) = 250 + 506*x + 135*x^2 + 2*x^3.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 10 + 5*x;
P[2, x_] := 50 + 52*x + 2*x^2;
P[3, x_] := 250 + 506*x + 135*x^2 + 2*x^3;
P[n_, x_] := P[n, x] = Expand[Together[((5*n - 15 + (5*n - 15)*x)*P[n - 2, x] + (5*n - 15)*x*D[P[n - 2, x], x] + (2*n - 6 + (n - 3)*x)*P[n - 1, x] + (n - 3)*x*D[P[n - 1, x], x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 10 + 5*x; v[3] = 50 + 52*x + 2*x^2; v[4] = 250 + 506*x + 135*x^2 + 2*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((5*n - 15 + (5*n - 15)*x)*v[i - 2] + (5*n - 15)*x*deriv(v[i - 2], x) + (2*n - 6 + (n - 3)*x)*v[i - 1] + (n - 3)*x*deriv(v[i - 1], x))/(n - 3)); 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