A084534 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(2 + 12\,x + 18\,x^{2} - n - 6\,n x - 9\,n x^{2}\right)\,P_{n-2}(x) + \left(-6 - 12\,x + 3\,n + 6\,n x\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 6\,x\)

\(P_{2}(x) = 7 + 24\,x + 18\,x^{2}\)

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	6														9
2	7	24	18													49
3	18	90	135	54												297
4	47	312	720	648	162											1889
5	123	1020	3195	4590	2835	486										12249
6	322	3204	12690	25164	25515	11664	1458									80017
7	843	9786	46872	118692	168399	129276	45927	4374								524169
8	2207	29280	164520	507168	926640	1006992	612360	174960	13122							3437249
9	5778	86238	556065	2020626	4520529	6372918	5541858	2755620	649539	39366						22548537
10	15127	250860	1825560	7640460	20235015	35110584	39869010	28605960	11908215	2361960	118098					147940849
11	39603	722436	5856345	27741582	84898935	175060116	245806407	231668910	140300424	49797990	8444007	354294				970691049
12	103682	2063304	18436788	97511256	338638482	809289144	1355132268	1586029896	1268549667	659931624	202656168	29760696	1062882			6369165857
13	271443	5851950	57142800	333772920	1296927450	3525570360	6860874150	9606834900	9579256830	6614472150	2997622485	806018850	103630995	3188646		41791435929
14	710647	16499112	174790350	1117473840	4803782760	14639554440	32477431140	52985323800	63397860435	54773458740	33100507440	13218709140	3143473515	357128352	9565938	274216269649

Technical details

Node id: n-c04eaa7a3fc005c1 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-2b7ed1e86e9f26f3

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,     6;
  2:    7,    24,     18;
  3:   18,    90,    135,      54;
  4:   47,   312,    720,     648,     162;
  5:  123,  1020,   3195,    4590,    2835,     486;
  6:  322,  3204,  12690,   25164,   25515,   11664,    1458;
  7:  843,  9786,  46872,  118692,  168399,  129276,   45927,    4374;
  8: 2207, 29280, 164520,  507168,  926640, 1006992,  612360,  174960,  13122;
  9: 5778, 86238, 556065, 2020626, 4520529, 6372918, 5541858, 2755620, 649539, 39366;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = -(n-2+(6*n-12)*x+(9*n-18)*x^2)*P(n-2,x) + (3*n-6+(6*n-12)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 6*x, P(2,x) = 7 + 24*x + 18*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[2, x_] := 7 + 24*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (6*n - 12)*x + (9*n - 18)*x^2)*P[n - 2, x] + (3*n - 6 + (6*n - 12)*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 + 6*x; v[3] = 7 + 24*x + 18*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 2 + (6*n - 12)*x + (9*n - 18)*x^2)*v[i - 2] + (3*n - 6 + (6*n - 12)*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 A013610

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