A102413 → basis A013610

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2 + 3\,x\)

\(P_{2}(x) = 6 + 18\,x + 9\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	6	18	9													33
3	14	63	81	27												185
4	34	204	414	324	81											1057
5	82	615	1710	2160	1215	243										6025
6	198	1782	6291	11016	9963	4374	729									34353
7	478	5019	21483	48195	60669	42525	15309	2187								195865
8	1154	13848	69660	190728	308934	301320	172044	52488	6561							1116737
9	2786	37611	217404	703080	1395306	1753974	1390932	669222	177147	19683						6367145
10	6726	100890	658845	2458080	5780970	8921988	9119790	6079860	2525985	590490	59049					36302673
11	16238	267927	1950597	8247987	22440726	41126778	51658398	44361108	25476363	9310059	1948617	177147				206981945
12	39202	705636	5666490	26782380	82780623	175840632	262582884	277136640	204801615	103217652	33657930	6377292	531441			1180120417
13	94642	1845519	16204266	84664008	293093073	708472089	1227726396	1541358234	1399572837	906579297	406847610	119751372	20726199	1594323		6728529865
14	228486	4798206	45728487	261722664	1003305933	2720405898	5370973839	7830780948	8459431911	6729499602	3876743997	1567396656	420369831	66961566	4782969	38363130993

Technical details

Node id: n-49a7d6c7beaa24e5 · revision 0 · JSON
Definition source: derived:basis_map_A013610
Fingerprint: f6-fd14f22a5134cfe6

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:    2,     3;
  2:    6,    18,      9;
  3:   14,    63,     81,     27;
  4:   34,   204,    414,    324,      81;
  5:   82,   615,   1710,   2160,    1215,     243;
  6:  198,  1782,   6291,  11016,    9963,    4374,     729;
  7:  478,  5019,  21483,  48195,   60669,   42525,   15309,   2187;
  8: 1154, 13848,  69660, 190728,  308934,  301320,  172044,  52488,   6561;
  9: 2786, 37611, 217404, 703080, 1395306, 1753974, 1390932, 669222, 177147, 19683;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(3*n-6)*x)*P(n-2,x) + (2*n-4+(3*n-6)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 3*x, P(2,x) = 6 + 18*x + 9*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[2, x_] := 6 + 18*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (3*n - 6)*x)*P[n - 2, x] + (2*n - 4 + (3*n - 6)*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] = 2 + 3*x; v[3] = 6 + 18*x + 9*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (3*n - 6)*x)*v[i - 2] + (2*n - 4 + (3*n - 6)*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,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Liu–Wang theorem with a lag row (above)

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