Real-rooted atlas

A133081 → reverse (shift +1) → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(6 - 10\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(12\,x + 6\,x^{2} - 38\,n x - 46\,n x^{2} + 32\,n^{2} x + 52\,n^{2} x^{2} - 8\,n^{3} x - 16\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(3\,x^{2} + 2\,x^{3} - 8\,n x^{2} - 12\,n x^{3} + 4\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1123
200044
300005611
4000000088
50000000091019
6000000000001212
7000000000000131427

All 15 rows

Technical details
Node id
n-ec4ecc8db13054c2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2cafe00cb2d17c6d

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
  • 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: 1, 2;
  2: 0, 0, 0, 4;
  3: 0, 0, 0, 0, 5, 6;
  4: 0, 0, 0, 0, 0, 0, 0, 8;
  5: 0, 0, 0, 0, 0, 0, 0, 0, 9, 10;
  6: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0, 12;
  7: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0, 13, 14;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0, 0, 16;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0, 0,  0, 17, 18;
%F Formula
The row polynomials satisfy (4*n^2-10*n+6)*P(n,x) = -((8*n^3-32*n^2+38*n-12)*x+(16*n^3-52*n^2+46*n-6)*x^2)*P(n-1,x) + ((4*n^2-8*n+3)*x^2+(8*n^2-12*n+2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((8*n^3 - 32*n^2 + 38*n - 12)*x + (16*n^3 - 52*n^2 + 46*n - 6)*x^2)*P[n - 1, x] + ((4*n^2 - 8*n + 3)*x^2 + (8*n^2 - 12*n + 2)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-((8*n^3 - 32*n^2 + 38*n - 12)*x + (16*n^3 - 52*n^2 + 46*n - 6)*x^2)*v[i - 1] + ((4*n^2 - 8*n + 3)*x^2 + (8*n^2 - 12*n + 2)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6)); 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.