Real-rooted atlas

Veronese parent of A109446 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
21438
336312
4181816548
55203020580
6112458075367256
7742105140105427448

All 15 rows

Technical details
Node id
n-b4f0fc8b093f0511 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-caa05c9bb4947825

Observed on the stored rows ?

  • Degrees: 0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14 (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,  4,   3;
  3: 3,  6,   3;
  4: 1,  8,  18,  16,   5;
  5: 5, 20,  30,  20,   5;
  6: 1, 12,  45,  80,  75,  36,   7;
  7: 7, 42, 105, 140, 105,  42,   7;
  8: 1, 16,  84, 224, 350, 336, 196, 64, 9;
  9: 9, 72, 252, 504, 630, 504, 252, 72, 9;
%F Formula
The row polynomials satisfy (n^2-n)*P(n,x) = -(n^3-3*n^2+2-(3*n^2-n-2)*x-(n^3-n)*x^2)*P(n-1,x) + (n^2-n-1+(n^2-3*n-2)*x-(n^2+3*n+1)*x^2-(n^2+n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 3*n^2 + 2 - (3*n^2 - n - 2)*x - (n^3 - n)*x^2)*P[n - 1, x] + (n^2 - n - 1 + (n^2 - 3*n - 2)*x - (n^2 + 3*n + 1)*x^2 - (n^2 + n)*x^3)*D[P[n - 1, x], x])/(n^2 - n)]];
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; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 3*n^2 + 2 - (3*n^2 - n - 2)*x - (n^3 - n)*x^2)*v[i - 1] + (n^2 - n - 1 + (n^2 - 3*n - 2)*x - (n^2 + 3*n + 1)*x^2 - (n^2 + n)*x^3)*deriv(v[i - 1], x))/(n^2 - n)); 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.