Real-rooted atlas

A133081: An interpolation operator, companion to A133080.

A133081 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A133081

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111
20112
30011
4000112
5000011
600000112
700000011
80000000112
90000000011
10000000000112
11000000000011
1200000000000112
1300000000000011
140000000000000112
Technical details
Node id
n-ddd13fb19530ec39 · revision 0 · JSON
Definition source
oeis_catalog:A133081
Fingerprint
f6-fea7ad1f3ec1f4c4

Observed on the stored rows ?

  • Degrees: 0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14 (irregular)
  • Palindromic rows: yes
  • γ-positive (gamma_positive): yes
  • 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: 0, 1, 1;
  3: 0, 0, 1;
  4: 0, 0, 0, 1, 1;
  5: 0, 0, 0, 0, 1;
  6: 0, 0, 0, 0, 0, 1, 1;
  7: 0, 0, 0, 0, 0, 0, 1;
  8: 0, 0, 0, 0, 0, 0, 0, 1, 1;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 1;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (n-2)*x^2*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2)*x^2*P[n - 2, 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] = 1; v[2] = 1; v[3] = x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2)*x^2*v[i - 2])/(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

Used in 1

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.