Real-rooted atlas

A107131 → basis A008277

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
22417
34169130
496363161152
521251400175251873
6511013243116303953615557
712741451446213951507577749138587

All 15 rows

Technical details
Node id
n-9302c5068cc7fd22 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-2bbe6df6dc024c8b

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:   1,     1;
  2:   2,     4,      1;
  3:   4,    16,      9,      1;
  4:   9,    63,     63,     16,      1;
  5:  21,   251,    400,    175,     25,      1;
  6:  51,  1013,   2431,   1630,    395,     36,     1;
  7: 127,  4145,  14462,  13951,   5075,    777,    49,    1;
  8: 323, 17181,  85163, 113638,  58065,  13174,  1386,   64,  1;
  9: 835, 72061, 499422, 897667, 618051, 194061, 29988, 2298, 81, 1;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (3*n-3+(n-1)*x-(n-1)*x^2)*P(n-2,x) + ((n-1)*x-(2*n-2)*x^2)*d/dx P(n-2,x) - (n-1)*x^2*d^2/dx^2 P(n-2,x) + (2*n+1+(2*n+1)*x)*P(n-1,x) + (2*n+1)*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (n - 1)*x - (n - 1)*x^2)*P[n - 2, x] + ((n - 1)*x - (2*n - 2)*x^2)*D[P[n - 2, x], x] - (n - 1)*x^2*D[P[n - 2, x], {x, 2}] + (2*n + 1 + (2*n + 1)*x)*P[n - 1, x] + (2*n + 1)*x*D[P[n - 1, x], x])/(n + 2)]];
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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (n - 1)*x - (n - 1)*x^2)*v[i - 2] + ((n - 1)*x - (2*n - 2)*x^2)*deriv(v[i - 2], x) - (n - 1)*x^2*deriv(deriv(v[i - 2], x), x) + (2*n + 1 + (2*n + 1)*x)*v[i - 1] + (2*n + 1)*x*deriv(v[i - 1], x))/(n + 2)); 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