A123097 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

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

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

\(P_{1}(x) = 5 + 2\,x\)

\(P_{2}(x) = 14 + 13\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	2							7
2	14	13	3						30
3	36	52	25	4					117
4	88	172	126	41	5				432
5	208	512	504	248	61	6			1539
6	480	1424	1760	1160	430	85	7		5346
7	1088	3776	5616	4640	2300	684	113	8	18225

All 15 rows

Technical details

Node id: n-ac1afc976a1e5fa6 · revision 0 · JSON
Definition source: derived:basis_map_A007318
Fingerprint: f6-b8552aefd92cb15e

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,     2;
  2:   14,    13,     3;
  3:   36,    52,    25,     4;
  4:   88,   172,   126,    41,     5;
  5:  208,   512,   504,   248,    61,     6;
  6:  480,  1424,  1760,  1160,   430,    85,    7;
  7: 1088,  3776,  5616,  4640,  2300,   684,  113,    8;
  8: 2432,  9664, 16800, 16688, 10360,  4116, 1022,  145,   9;
  9: 5376, 24064, 47872, 55552, 41440, 20608, 6832, 1456, 181, 10;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(4*n-8)*x+(n-2)*x^2)*P(n-2,x) + (4*n-8+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 2*x, P(2,x) = 14 + 13*x + 3*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 2*x;
P[2, x_] := 14 + 13*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*P[n - 2, x] + (4*n - 8 + (2*n - 4)*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 + 2*x; v[3] = 14 + 13*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*v[i - 2] + (4*n - 8 + (2*n - 4)*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