Real-rooted atlas

A133087 → reverse → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(-2 + 4\,x - 5\,n - 10\,n x + 6\,n x^{2} + 3\,n^{2} - 9\,n^{2} x^{2} + 2\,n^{3} x + 3\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(2\,x + 2\,x^{2} - 2\,n - 4\,n x + n x^{2} + 3\,n x^{3} - 2\,n^{2} x - 5\,n^{2} x^{2} - 3\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
2224
3410620
441212432
5628483610128
66306060306192
78541502201807814704

All 15 rows

Technical details
Node id
n-e254a6c04eed3447 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-c2718fb628e717d5

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, 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:  2;
  2:  2,  2;
  3:  4, 10,   6;
  4:  4, 12,  12,   4;
  5:  6, 28,  48,  36,  10;
  6:  6, 30,  60,  60,  30,   6;
  7:  8, 54, 150, 220, 180,  78,  14;
  8:  8, 56, 168, 280, 280, 168,  56,   8;
  9: 10, 88, 336, 728, 980, 840, 448, 136, 18;
%F Formula
The row polynomials satisfy (n^2-3*n+2)*P(n,x) = (3*n^2-5*n-2+(2*n^3-10*n+4)*x+(3*n^3-9*n^2+6*n)*x^2)*P(n-1,x) - (2*n+(2*n^2+4*n-2)*x+(5*n^2-n-2)*x^2+(3*n^2-3*n)*x^3)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 2, P(2,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n^2 - 5*n - 2 + (2*n^3 - 10*n + 4)*x + (3*n^3 - 9*n^2 + 6*n)*x^2)*P[n - 1, x] - (2*n + (2*n^2 + 4*n - 2)*x + (5*n^2 - n - 2)*x^2 + (3*n^2 - 3*n)*x^3)*D[P[n - 1, x], x])/(n^2 - 3*n + 2)]];
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] = 2; v[3] = 2 + 2*x; for(n = 3, N, my(i = n + 1); v[i] = ((3*n^2 - 5*n - 2 + (2*n^3 - 10*n + 4)*x + (3*n^3 - 9*n^2 + 6*n)*x^2)*v[i - 1] - (2*n + (2*n^2 + 4*n - 2)*x + (5*n^2 - n - 2)*x^2 + (3*n^2 - 3*n)*x^3)*deriv(v[i - 1], x))/(n^2 - 3*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

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.