A123097 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 16\,x + 8\,x^{2} - 4\,n - 8\,n x - 4\,n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 8\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 4\,x\)

\(P_{2}(x) = 14 + 26\,x + 12\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	4							9
2	14	26	12						52
3	36	104	100	32					272
4	88	344	504	328	80				1344
5	208	1024	2016	1984	976	192			6400
6	480	2848	7040	9280	6880	2720	448		29696
7	1088	7552	22464	37120	36800	21888	7232	1024	135168

All 15 rows

Technical details

Node id: n-3d01f9226121973c · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-36dd966291f10a09

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:    1;
  1:    5,     4;
  2:   14,    26,     12;
  3:   36,   104,    100,     32;
  4:   88,   344,    504,    328,     80;
  5:  208,  1024,   2016,   1984,    976,    192;
  6:  480,  2848,   7040,   9280,   6880,   2720,    448;
  7: 1088,  7552,  22464,  37120,  36800,  21888,   7232,   1024;
  8: 2432, 19328,  67200, 133504, 165760, 131712,  65408,  18560,  2304;
  9: 5376, 48128, 191488, 444416, 663040, 659456, 437248, 186368, 46336, 5120;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(8*n-16)*x+(4*n-8)*x^2)*P(n-2,x) + (4*n-8+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 4*x, P(2,x) = 14 + 26*x + 12*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[2, x_] := 14 + 26*x + 12*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*P[n - 2, x] + (4*n - 8 + (4*n - 8)*x)*P[n - 1, 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] = 5 + 4*x; v[3] = 14 + 26*x + 12*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (8*n - 16)*x + (4*n - 8)*x^2)*v[i - 2] + (4*n - 8 + (4*n - 8)*x)*v[i - 1])/(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

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

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows: T(n,k) = binomial(n-2, k-1) + n*binomial(n-1, k-1), 1 <= k <= n, starting with T(1, 1) = 1.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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