Real-rooted atlas

A266326: Triangle read by rows giving successive states of cellular automaton generated by "Rule 20" initiated with a single ON (black) cell.

A133081 → reverse (shift +1)

A266326 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A266326

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
10112
2000011
300000112
40000000011
5000000000112
600000000000011
70000000000000112
8000000000000000011
900000000000000000112
100000000000000000000011
11000000000000000000000112
1200000000000000000000000011
130000000000000000000000000112
14000000000000000000000000000011
Technical details
Node id
n-73781d44d680ea54 · revision 0 · JSON
Definition source
oeis_catalog:A266326
Fingerprint
f6-1ba04bc9f002fd9b

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, intercept 0)
  • 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: 0, 1, 1;
  2: 0, 0, 0, 0, 1;
  3: 0, 0, 0, 0, 0, 1, 1;
  4: 0, 0, 0, 0, 0, 0, 0, 0, 1;
  5: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
  6: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  7: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
%F Formula
The row polynomials satisfy P(n,x) = -((2*n-3)*x+(4*n-5)*x^2)*P(n-1,x) + (x^2+2*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-((2*n - 3)*x + (4*n - 5)*x^2)*P[n - 1, x] + (x^2 + 2*x^3)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -((2*n - 3)*x + (4*n - 5)*x^2)*v[i - 1] + (x^2 + 2*x^3)*deriv(v[i - 1], x)); 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.