A102413 → basis A013609

nonnegative coefficients

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

Recurrence

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

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

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

\(P_{2}(x) = 6 + 12\,x + 4\,x^{2}\)

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	2														4
2	6	12	4													22
3	14	42	36	8												100
4	34	136	184	96	16											466
5	82	410	760	640	240	32										2164
6	198	1188	2796	3264	1968	576	64									10054
7	478	3346	9548	14280	11984	5600	1344	128								46708
8	1154	9232	30960	56512	61024	39680	15104	3072	256							216994
9	2786	25074	96624	208320	275616	230976	122112	39168	6912	512						1008100
10	6726	67260	292820	728320	1141920	1174912	800640	355840	98560	15360	1024					4683382
11	16238	178618	866932	2443848	4432736	5415872	4535168	2596352	994048	242176	33792	2048				21757828
12	39202	470424	2518440	7935520	16351728	23155968	23052544	16220160	7991040	2684928	583680	73728	4096			101081458
13	94642	1230346	7201896	25085632	57894928	93296736	107783936	90212096	54609152	23582208	7055360	1384448	159744	8192		469599316
14	228486	3198804	20323772	77547456	198183888	358242752	471525824	458317312	330073856	175049728	67228672	18120704	3239936	344064	16384	2181641638

Technical details

Node id: n-316742a6512c57a1 · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-22e87c6bce817021

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,     2;
  2:    6,    12,     4;
  3:   14,    42,    36,      8;
  4:   34,   136,   184,     96,     16;
  5:   82,   410,   760,    640,    240,     32;
  6:  198,  1188,  2796,   3264,   1968,    576,     64;
  7:  478,  3346,  9548,  14280,  11984,   5600,   1344,   128;
  8: 1154,  9232, 30960,  56512,  61024,  39680,  15104,  3072,  256;
  9: 2786, 25074, 96624, 208320, 275616, 230976, 122112, 39168, 6912, 512;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(2*n-4)*x)*P(n-2,x) + (2*n-4+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 6 + 12*x + 4*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 6 + 12*x + 4*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (2*n - 4)*x)*P[n - 2, x] + (2*n - 4 + (2*n - 4)*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 + 2*x; v[3] = 6 + 12*x + 4*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (2*n - 4)*x)*v[i - 2] + (2*n - 4 + (2*n - 4)*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