A206306 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 4 + 20\,x + 16\,x^{2}\)

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	4														5
2	4	20	16													40
3	14	88	144	64												310
4	48	368	896	832	256											2400
5	164	1488	4768	6784	4352	1024										18580
6	560	5872	23232	44544	44032	21504	4096									143840
7	1912	22752	106880	257536	345600	260096	102400	16384								1113560
8	6528	86912	472064	1368576	2324480	2379776	1441792	475136	65536							8620800
9	22288	328256	2022144	6847488	14081024	18296832	15081472	7634944	2162688	262144						66739280
10	76096	1228352	8457472	32741376	79065088	124751872	130629632	89915392	39059456	9699328	1048576					516672640
11	259808	4561280	34699008	151100416	419063808	778674176	991363072	866910208	511574016	194510848	42991616	4194304				3999902560
12	887040	16827648	140126208	677714944	2122526720	4541448192	6818889728	7253262336	5435817984	2804940800	947912704	188743680	16777216			30965875200
13	3028544	61736192	558417408	2969163776	10362839040	25098551296	43458625536	54554787840	49733173248	32574013440	14925430784	4538236928	822083584	67108864		239727196480
14	10340096	225403648	2200361984	12754894848	49082957824	132762664960	260590927872	377547128832	406280208384	323618865152	188101951488	77477183488	21407727616	3556769792	268435456	1855885821440

Technical details

Node id: n-cc618c8c4d7fd590 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-ed2231ad672435a3

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,      4;
  2:     4,     20,      16;
  3:    14,     88,     144,      64;
  4:    48,    368,     896,     832,      256;
  5:   164,   1488,    4768,    6784,     4352,     1024;
  6:   560,   5872,   23232,   44544,    44032,    21504,     4096;
  7:  1912,  22752,  106880,  257536,   345600,   260096,   102400,   16384;
  8:  6528,  86912,  472064, 1368576,  2324480,  2379776,  1441792,  475136,   65536;
  9: 22288, 328256, 2022144, 6847488, 14081024, 18296832, 15081472, 7634944, 2162688, 262144;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(2*n-4)*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) = 1 + 4*x, P(2,x) = 4 + 20*x + 16*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 4 + 20*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4)*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] = 1 + 4*x; v[3] = 4 + 20*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(2*n - 4)*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)
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