Real-rooted atlas

A152828 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-60 + 47\,n - 12\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(432 + 216\,x - 104\,n - 52\,n x - 6\,n^{2} - 3\,n^{2} x + 2\,n^{3} + n^{3} x\right)\,P_{n-3}(x) + \left(756 + 756\,x + 189\,x^{2} - 108\,n - 108\,n x - 27\,n x^{2}\right)\,P_{n-3}'(x) + \left(216 + 324\,x + 162\,x^{2} + 27\,x^{3}\right)\,P_{n-3}''(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 1\)
\(P_{4}(x) = 1\)
\(P_{5}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
00
10
20
311
411
511
6426
7426
8426
91212327
101212327
111212327
123248244108
133248244108
143248244108
Technical details
Node id
n-7bc0dafb54ade7ab · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-19c649f255ec1907

Observed on the stored rows ?

  • Degrees: -1, -1, -1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3 (irregular)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  0;
  1:  0;
  2:  0;
  3:  1;
  4:  1;
  5:  1;
  6:  4,  2;
  7:  4,  2;
  8:  4,  2;
  9: 12, 12, 3;
%F Formula
The row polynomials satisfy (n^3-12*n^2+47*n-60)*P(n,x) = (2*n^3-6*n^2-104*n+432+(n^3-3*n^2-52*n+216)*x)*P(n-3,x) - (108*n-756+(108*n-756)*x+(27*n-189)*x^2)*d/dx P(n-3,x) + (216+324*x+162*x^2+27*x^3)*d^2/dx^2 P(n-3,x) for n >= 6, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 1, P(4,x) = 1, P(5,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 1;
P[4, x_] := 1;
P[5, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 6*n^2 - 104*n + 432 + (n^3 - 3*n^2 - 52*n + 216)*x)*P[n - 3, x] - (108*n - 756 + (108*n - 756)*x + (27*n - 189)*x^2)*D[P[n - 3, x], x] + (216 + 324*x + 162*x^2 + 27*x^3)*D[P[n - 3, x], {x, 2}])/(n^3 - 12*n^2 + 47*n - 60)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 0; v[4] = 1; v[5] = 1; v[6] = 1; for(n = 6, N, my(i = n + 1); v[i] = ((2*n^3 - 6*n^2 - 104*n + 432 + (n^3 - 3*n^2 - 52*n + 216)*x)*v[i - 3] - (108*n - 756 + (108*n - 756)*x + (27*n - 189)*x^2)*deriv(v[i - 3], x) + (216 + 324*x + 162*x^2 + 27*x^3)*deriv(deriv(v[i - 3], x), x))/(n^3 - 12*n^2 + 47*n - 60)); 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

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.