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	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	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
8	2207	19520	73120	150272	183040	132608	53760	10240	512							625279
9	5778	57492	247140	598704	892944	839232	486528	161280	25344	1024						3315466
10	15127	167240	811360	2263840	3997040	4623616	3500160	1674240	464640	61440	2048					17580751
11	39603	481624	2602820	8219728	16770160	23053184	21579712	13559040	5474304	1295360	146432	4096				93226063
12	103682	1375536	8194128	28892224	66891552	106573056	118969088	92826624	49496832	17166336	3514368	344064	8192			494355682
13	271443	3901300	25396800	98895680	256183200	464272640	602326400	562265600	373767680	172057600	51983360	9318400	798720	16384		2621455207
14	710647	10999408	77684600	331103360	948895360	1927842560	2851242240	3101107200	2473685760	1424783360	574013440	152821760	24227840	1835008	32768	13900985311

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