A267006: Triangle read by rows giving successive states of cellular automaton generated by "Rule 84" initiated with a single ON (black) cell.

A267006 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A267006

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

\(P_{0}(x) = 1\)

\(P_{1}(x) = x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	0	1	1																											2
2	0	0	0	1	1																									2
3	0	0	0	0	0	1	1																							2
4	0	0	0	0	0	0	0	1	1																					2
5	0	0	0	0	0	0	0	0	0	1	1																			2
6	0	0	0	0	0	0	0	0	0	0	0	1	1																	2
7	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1															2
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1													2
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1											2
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1									2
11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1							2
12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1					2
13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1			2
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	2

Technical details

Node id: n-e1c721301bf80375 · revision 1 · JSON
Definition source: oeis_catalog:A267006
Fingerprint: f6-abab48ed12ce8f73

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, 1, 1;
  3: 0, 0, 0, 0, 0, 1, 1;
  4: 0, 0, 0, 0, 0, 0, 0, 1, 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, 1, 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, 1, 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 (n-1)*P(n,x) = (n-1)*x^2*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1)*x^2*P[n - 1, x])/(n - 1)]];
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] = x + x^2; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1)*x^2*v[i - 1])/(n - 1)); 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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers

Proof chains ? JSON

Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Related ?

Same fingerprint ?

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	0	1	1																											2
2	0	0	0	1	1																									2
3	0	0	0	0	0	1	1																							2
4	0	0	0	0	0	0	0	1	1																					2
5	0	0	0	0	0	0	0	0	0	1	1																			2
6	0	0	0	0	0	0	0	0	0	0	0	1	1																	2
7	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1															2
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1													2
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1											2
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1									2
11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1							2
12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1					2
13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1			2
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	2

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	0	1	1																											2
2	0	0	0	1	1																									2
3	0	0	0	0	0	1	1																							2
4	0	0	0	0	0	0	0	1	1																					2
5	0	0	0	0	0	0	0	0	0	1	1																			2
6	0	0	0	0	0	0	0	0	0	0	0	1	1																	2
7	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1															2
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1													2
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1											2
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1									2
11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1							2
12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1					2
13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1			2
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	2

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	0	1	1																											2
2	0	0	0	1	1																									2
3	0	0	0	0	0	1	1																							2
4	0	0	0	0	0	0	0	1	1																					2
5	0	0	0	0	0	0	0	0	0	1	1																			2
6	0	0	0	0	0	0	0	0	0	0	0	1	1																	2
7	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1															2
8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1													2
9	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1											2
10	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1									2
11	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1							2
12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1					2
13	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1			2
14	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	2