A206306 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 4 + 10\,x + 4\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	4	10	4													18
3	14	44	36	8												102
4	48	184	224	104	16											576
5	164	744	1192	848	272	32										3252
6	560	2936	5808	5568	2752	672	64									18360
7	1912	11376	26720	32192	21600	8128	1600	128								103656
8	6528	43456	118016	171072	145280	74368	22528	3712	256							585216
9	22288	164128	505536	855936	880064	571776	235648	59648	8448	512						3303984
10	76096	614176	2114368	4092672	4941568	3898496	2041088	702464	152576	18944	1024					18653472
11	259808	2280640	8674752	18887552	26191488	24333568	15490048	6772736	1998336	379904	41984	2048				105312864
12	887040	8413824	35031552	84714368	132657920	141920256	106545152	56666112	21233664	5478400	925696	92160	4096			594570240
13	3028544	30868096	139604352	371145472	647677440	784329728	679041024	426209280	194270208	63621120	14575616	2215936	200704	8192		3356795712
14	10340096	112701824	550090496	1594361856	3067684864	4148833280	4071733248	2949586944	1587032064	632068096	183693312	37830656	5226496	434176	16384	18951633792

Technical details

Node id: n-a8289b7270086fb2 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-9bfdbf18d27088d9

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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:     1;
  1:     1,      2;
  2:     4,     10,      4;
  3:    14,     44,     36,      8;
  4:    48,    184,    224,    104,     16;
  5:   164,    744,   1192,    848,    272,     32;
  6:   560,   2936,   5808,   5568,   2752,    672,     64;
  7:  1912,  11376,  26720,  32192,  21600,   8128,   1600,   128;
  8:  6528,  43456, 118016, 171072, 145280,  74368,  22528,  3712,  256;
  9: 22288, 164128, 505536, 855936, 880064, 571776, 235648, 59648, 8448, 512;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(2*n-4)*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) = 1 + 2*x, P(2,x) = 4 + 10*x + 4*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 4 + 10*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4)*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] = 1 + 2*x; v[3] = 4 + 10*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(2*n - 4)*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)
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 Riordan array (1, x/(1-3*x+2*x^2)).?
  - 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