A128908 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 - n\right)\,P_{n-2}(x) + \left(-6 - 4\,x + 3\,n + 2\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 3 + 8\,x + 4\,x^{2}\)

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	3	8	4													15
3	8	28	28	8												72
4	21	92	136	80	16											345
5	55	290	564	504	208	32										1653
6	144	888	2136	2560	1616	512	64									7920
7	377	2662	7620	11448	9760	4736	1216	128								37947
8	987	7852	26048	47024	50560	33216	13056	2816	256							181815
9	2584	22868	86228	181720	235968	196032	104384	34432	6400	512						871128
10	6765	65920	278372	670592	1020784	1026816	692160	309248	87808	14336	1024					4173825
11	17711	188422	880728	2386800	4167568	4925984	4025728	2277632	875520	218112	31744	2048				19997997
12	46368	534768	2740656	8251264	16255520	22086272	21236992	14575104	7094016	2391040	530432	69632	4096			95816160
13	121393	1508618	8410776	27848304	61101520	93843872	103857792	83921664	49556736	21143040	6341632	1267712	151552	8192		459082803
14	317811	4233872	25508908	92115200	222745648	381648384	478024128	444905472	309419520	160151552	60780544	16416768	2985984	327680	16384	2199597855

Technical details

Node id: n-c9e16333e866e154 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-47d3e893e08f85d1

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:    3,     8,     4;
  3:    8,    28,    28,      8;
  4:   21,    92,   136,     80,     16;
  5:   55,   290,   564,    504,    208,     32;
  6:  144,   888,  2136,   2560,   1616,    512,     64;
  7:  377,  2662,  7620,  11448,   9760,   4736,   1216,   128;
  8:  987,  7852, 26048,  47024,  50560,  33216,  13056,  2816,  256;
  9: 2584, 22868, 86228, 181720, 235968, 196032, 104384, 34432, 6400, 512;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(n-2)*P(n-2,x) + (3*n-6+(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) = 3 + 8*x + 4*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[2, x_] := 3 + 8*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2)*P[n - 2, x] + (3*n - 6 + (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] = 3 + 8*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 2)*v[i - 2] + (3*n - 6 + (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-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