A206306: Riordan array (1, x/(1-3x+2x^2)).

A206306 ↗ nonnegative coefficients

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

Recurrence from A206306

\(\left(-2 + n\right)\,P_{n}(x) = \left(-6 - 2\,x + 3\,n + n x\right)\,P_{n-1}(x) + \left(4 - 2\,n\right)\,P_{n-2}(x)\)

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

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

\(P_{2}(x) = 3\,x + x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	1							1
2	0	3	1						4
3	0	7	6	1					14
4	0	15	23	9	1				48
5	0	31	72	48	12	1			164
6	0	63	201	198	82	15	1		560
7	0	127	522	699	420	125	18	1	1912

All 15 rows b-files: A007070 (row sums), A110441, A204089 (row sums), A206306 ?

Technical details

Node id: n-d7abbe24afd98bff · revision 0 · JSON
Definition source: oeis_catalog:A206306
Fingerprint: f6-37c3559a5d3b5b73

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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): no

Matches ?

A206306 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A110441 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A204089, A007070

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0: 1;
  1: 0,   1;
  2: 0,   3,    1;
  3: 0,   7,    6,    1;
  4: 0,  15,   23,    9,    1;
  5: 0,  31,   72,   48,   12,    1;
  6: 0,  63,  201,  198,   82,   15,    1;
  7: 0, 127,  522,  699,  420,  125,   18,   1;
  8: 0, 255, 1291, 2223, 1795,  765,  177,  21,  1;
  9: 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(n-2)*x)*P(n-1,x) - (2*n-4)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 3*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[2, x_] := 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (n - 2)*x)*P[n - 1, x] - (2*n - 4)*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] = x; v[3] = 3*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n - 6 + (n - 2)*x)*v[i - 1] - (2*n - 4)*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

A110441 triangle, offset 0, from P₁, low zeros dropped

%e Example

Triangle begins:
  0:    1;
  1:    3,    1;
  2:    7,    6,     1;
  3:   15,   23,     9,     1;
  4:   31,   72,    48,    12,     1;
  5:   63,  201,   198,    82,    15,    1;
  6:  127,  522,   699,   420,   125,   18,    1;
  7:  255, 1291,  2223,  1795,   765,  177,   21,   1;
  8:  511, 3084,  6562,  6768,  3840, 1260,  238,  24,  1;
  9: 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where (n-1)*P(n,x) = (3*n-3+(n-1)*x)*P(n-1,x) - (2*n-2)*P(n-2,x) for n >= 2, with P(0,x) = x, P(1,x) = 3*x + x^2.

%t Mathematica

P[0, x_] := x;
P[1, x_] := 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 3 + (n - 1)*x)*P[n - 1, x] - (2*n - 2)*P[n - 2, x])/(n - 1)]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = 3*x + x^2; for(n = 2, N, my(i = n + 1); v[i] = ((3*n - 3 + (n - 1)*x)*v[i - 1] - (2*n - 2)*v[i - 2])/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A206306 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0,   1;
  2: 0,   3,    1;
  3: 0,   7,    6,    1;
  4: 0,  15,   23,    9,    1;
  5: 0,  31,   72,   48,   12,    1;
  6: 0,  63,  201,  198,   82,   15,    1;
  7: 0, 127,  522,  699,  420,  125,   18,   1;
  8: 0, 255, 1291, 2223, 1795,  765,  177,  21,  1;
  9: 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (3*n-6+(n-2)*x)*P(n-1,x) - (2*n-4)*P(n-2,x) for n >= 3, with P(0,x) = 1, P(1,x) = x, P(2,x) = 3*x + x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[2, x_] := 3*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n - 6 + (n - 2)*x)*P[n - 1, x] - (2*n - 4)*P[n - 2, x])/(n - 2)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = x; v[3] = 3*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = ((3*n - 6 + (n - 2)*x)*v[i - 1] - (2*n - 4)*v[i - 2])/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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 Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

Used in 6

Related ?

Same fingerprint ?

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003