A084534 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 3 + 4\,x\)

\(P_{2}(x) = 7 + 16\,x + 8\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	4							7
2	7	16	8						31
3	18	60	60	16					154
4	47	208	320	192	32				799
5	123	680	1420	1360	560	64			4207
6	322	2136	5640	7456	5040	1536	128		22258
7	843	6524	20832	35168	33264	17024	4032	256	117943

All 15 rows

Technical details

Node id: n-12a33c74cd3c78c7 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-b860ced141a23493

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:    3,     4;
  2:    7,    16,      8;
  3:   18,    60,     60,     16;
  4:   47,   208,    320,    192,     32;
  5:  123,   680,   1420,   1360,    560,     64;
  6:  322,  2136,   5640,   7456,   5040,   1536,    128;
  7:  843,  6524,  20832,  35168,  33264,  17024,   4032,    256;
  8: 2207, 19520,  73120, 150272, 183040, 132608,  53760,  10240,   512;
  9: 5778, 57492, 247140, 598704, 892944, 839232, 486528, 161280, 25344, 1024;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(n-2+(4*n-8)*x+(4*n-8)*x^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) = 3 + 4*x, P(2,x) = 7 + 16*x + 8*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[2, x_] := 7 + 16*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (4*n - 8)*x + (4*n - 8)*x^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] = 3 + 4*x; v[3] = 7 + 16*x + 8*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 2 + (4*n - 8)*x + (4*n - 8)*x^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 Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

Related ?

Same fingerprint ?

A086645 → basis A013611 → basis A013609

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023