Real-rooted atlas

A274498 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-24 + 9\,n - 6\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-18 - 12\,x - 6\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 6\)

Rows ?

nx0x1x2Pn(1)
00
10
266
31818
410872180
5324216540
6145819446484050
743745832194412150

All 15 rows

Technical details
Node id
n-435fc527bdda0c81 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-e309ee14657f718e

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
  • 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:     6;
  3:    18;
  4:   108,     72;
  5:   324,    216;
  6:  1458,   1944,   648;
  7:  4374,   5832,  1944;
  8: 17496,  34992, 23328,  5184;
  9: 52488, 104976, 69984, 15552;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (9*n-24+(2*n^2-6*n)*x)*P(n-1,x) - (18+(6*n+12)*x+4*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - 24 + (2*n^2 - 6*n)*x)*P[n - 1, x] - (18 + (6*n + 12)*x + 4*n*x^2)*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] = 0; v[2] = 0; v[3] = 6; for(n = 3, N, my(i = n + 1); v[i] = ((9*n - 24 + (2*n^2 - 6*n)*x)*v[i - 1] - (18 + (6*n + 12)*x + 4*n*x^2)*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

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.