Real-rooted atlas

A152828 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-48 - 156\,x - 27\,x^{2} + 27\,x^{3} - 43\,n + 9\,n x + 27\,n x^{2} + 6\,n^{2} + 9\,n^{2} x + n^{3}\right)\,P_{n}(x) = \left(-84 - 516\,x - 237\,x^{2} + 54\,x^{3} + 27\,x^{4} - 176\,n - 16\,n x + 90\,n x^{2} + 27\,n x^{3} + 30\,n^{2} + 33\,n^{2} x + 9\,n^{2} x^{2} + 2\,n^{3} + n^{3} x\right)\,P_{n-3}(x) + \left(540 + 432\,x + 27\,x^{2} - 27\,x^{3} - 108\,n - 108\,n x - 27\,n x^{2}\right)\,P_{n-3}'(x) + \left(216 + 324\,x + 162\,x^{2} + 27\,x^{3}\right)\,P_{n-3}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
111
211
3314
4314
5314
686115
786115
886115
920249154
1020249154
1120249154
12488048121189
13488048121189
14488048121189
Technical details
Node id
n-7ebe4b47815ec3b3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-481cf546ca14f6fd

Observed on the stored rows ?

  • Degrees: 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4 (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:  1;
  1:  1;
  2:  1;
  3:  3,  1;
  4:  3,  1;
  5:  3,  1;
  6:  8,  6, 1;
  7:  8,  6, 1;
  8:  8,  6, 1;
  9: 20, 24, 9, 1;
%F Formula
The row polynomials satisfy (n^3+6*n^2-43*n-48+(9*n^2+9*n-156)*x+(27*n-27)*x^2+27*x^3)*P(n,x) = (2*n^3+30*n^2-176*n-84+(n^3+33*n^2-16*n-516)*x+(9*n^2+90*n-237)*x^2+(27*n+54)*x^3+27*x^4)*P(n-3,x) - (108*n-540+(108*n-432)*x+(27*n-27)*x^2+27*x^3)*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 >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 30*n^2 - 176*n - 84 + (n^3 + 33*n^2 - 16*n - 516)*x + (9*n^2 + 90*n - 237)*x^2 + (27*n + 54)*x^3 + 27*x^4)*P[n - 3, x] - (108*n - 540 + (108*n - 432)*x + (27*n - 27)*x^2 + 27*x^3)*D[P[n - 3, x], x] + (216 + 324*x + 162*x^2 + 27*x^3)*D[P[n - 3, x], {x, 2}])/(n^3 + 6*n^2 - 43*n - 48 + (9*n^2 + 9*n - 156)*x + (27*n - 27)*x^2 + 27*x^3)]];
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; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^3 + 30*n^2 - 176*n - 84 + (n^3 + 33*n^2 - 16*n - 516)*x + (9*n^2 + 90*n - 237)*x^2 + (27*n + 54)*x^3 + 27*x^4)*v[i - 3] - (108*n - 540 + (108*n - 432)*x + (27*n - 27)*x^2 + 27*x^3)*deriv(v[i - 3], x) + (216 + 324*x + 162*x^2 + 27*x^3)*deriv(deriv(v[i - 3], x), x))/(n^3 + 6*n^2 - 43*n - 48 + (9*n^2 + 9*n - 156)*x + (27*n - 27)*x^2 + 27*x^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

Proof chains ? JSON

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.