A123097 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(8 + 32\,x + 32\,x^{2} - 4\,n - 16\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-8 - 16\,x + 4\,n + 8\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 5 + 8\,x\)

\(P_{2}(x) = 14 + 52\,x + 48\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	5	8							13
2	14	52	48						114
3	36	208	400	256					900
4	88	688	2016	2624	1280				6696
5	208	2048	8064	15872	15616	6144			47952
6	480	5696	28160	74240	110080	87040	28672		334368
7	1088	15104	89856	296960	588800	700416	462848	131072	2286144

All 15 rows

Technical details

Node id: n-e1c86e59a23114f8 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-b7833af94757c5d5

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,     8;
  2:   14,    52,     48;
  3:   36,   208,    400,     256;
  4:   88,   688,   2016,    2624,     1280;
  5:  208,  2048,   8064,   15872,    15616,     6144;
  6:  480,  5696,  28160,   74240,   110080,    87040,    28672;
  7: 1088, 15104,  89856,  296960,   588800,   700416,   462848,   131072;
  8: 2432, 38656, 268800, 1068032,  2652160,  4214784,  4186112,  2375680,   589824;
  9: 5376, 96256, 765952, 3555328, 10608640, 21102592, 27983872, 23855104, 11862016, 2621440;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(16*n-32)*x+(16*n-32)*x^2)*P(n-2,x) + (4*n-8+(8*n-16)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 5 + 8*x, P(2,x) = 14 + 52*x + 48*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 5 + 8*x;
P[2, x_] := 14 + 52*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (16*n - 32)*x + (16*n - 32)*x^2)*P[n - 2, x] + (4*n - 8 + (8*n - 16)*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 + 8*x; v[3] = 14 + 52*x + 48*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (16*n - 32)*x + (16*n - 32)*x^2)*v[i - 2] + (4*n - 8 + (8*n - 16)*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