A201730 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2								2
3	6	4							10
4	20	24							44
5	68	112	16						196
6	232	480	160						872
7	792	1968	1056	64					3880
8	2704	7840	5824	896					17264
9	9232	30592	29056	7680	256				76816
10	31520	117504	135936	52224	4608				341792
11	107616	445760	608000	309760	48640	1024			1520800
12	367424	1674112	2630144	1678336	394240	22528			6766784
13	1254464	6235392	11087616	8525824	2718720	282624	4096		30108736
14	4283008	23063040	45786624	41267200	16799744	2662400	106496		133968512
15	14623104	84799232	185912832	192367616	95864832	20959232	1548288	16384	596091520

Technical details

Node id: n-5a4f99377e2c235f · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-85d765f507ec68c6

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept -1)
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 ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     1;
   2:     2;
   3:     6,      4;
   4:    20,     24;
   5:    68,    112,     16;
   6:   232,    480,    160;
   7:   792,   1968,   1056,    64;
   8:  2704,   7840,   5824,   896;
   9:  9232,  30592,  29056,  7680,  256;
  10: 31520, 117504, 135936, 52224, 4608;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(2 - 4*x)*P[n - 2, x] + 4*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = -(2 - 4*x)*v[i - 2] + 4*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA …?
  - 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