A128908 → basis A013611

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 - n\right)\,P_{n-2}(x) + \left(-6 - 8\,x + 3\,n + 4\,n x\right)\,P_{n-1}(x)\)

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

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

\(P_{2}(x) = 3 + 16\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	4							5
2	3	16	16						35
3	8	56	112	64					240
4	21	184	544	640	256				1645
5	55	580	2256	4032	3328	1024			11275
6	144	1776	8544	20480	25856	16384	4096		77280
7	377	5324	30480	91584	156160	151552	77824	16384	529685

All 15 rows

Technical details

Node id: n-dbecd7b06e6b1292 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-4465f9855f0257a8

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:    3,    16,     16;
  3:    8,    56,    112,      64;
  4:   21,   184,    544,     640,     256;
  5:   55,   580,   2256,    4032,    3328,    1024;
  6:  144,  1776,   8544,   20480,   25856,   16384,    4096;
  7:  377,  5324,  30480,   91584,  156160,  151552,   77824,   16384;
  8:  987, 15704, 104192,  376192,  808960, 1062912,  835584,  360448,   65536;
  9: 2584, 45736, 344912, 1453760, 3775488, 6273024, 6680576, 4407296, 1638400, 262144;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(n-2)*P(n-2,x) + (3*n-6+(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) = 3 + 16*x + 16*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[2, x_] := 3 + 16*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2)*P[n - 2, x] + (3*n - 6 + (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] = 3 + 16*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 2)*v[i - 2] + (3*n - 6 + (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-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