A102413 → basis A013611

nonnegative coefficients

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

Recurrence

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

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

\(P_{1}(x) = 2 + 4\,x\)

\(P_{2}(x) = 6 + 24\,x + 16\,x^{2}\)

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	4														6
2	6	24	16													46
3	14	84	144	64												306
4	34	272	736	768	256											2066
5	82	820	3040	5120	3840	1024										13926
6	198	2376	11184	26112	31488	18432	4096									93886
7	478	6692	38192	114240	191744	179200	86016	16384								632946
8	1154	18464	123840	452096	976384	1269760	966656	393216	65536							4267106
9	2786	50148	386496	1666560	4409856	7391232	7815168	5013504	1769472	262144						28767366
10	6726	134520	1171280	5826560	18270720	37597184	51240960	45547520	25231360	7864320	1048576					193939726
11	16238	357236	3467728	19550784	70923776	173307904	290250752	332333056	254476288	123994112	34603008	4194304				1307475186
12	39202	940848	10073760	63484160	261627648	740990976	1475362816	2076180480	2045706240	1374683136	597688320	150994944	16777216			8814549746
13	94642	2460692	28807584	200685056	926318848	2985495552	6898171904	11547148288	13979942912	12074090496	7224688640	2835349504	654311424	67108864		59424674406
14	228486	6397608	81295088	620379648	3170942208	11463768064	30177652736	58664615936	84498907136	89625460736	68842160128	37111201792	13270777856	2818572288	268435456	400620795166

Technical details

Node id: n-22be1063864c8769 · revision 0 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-c64000922872a417

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,     4;
  2:    6,    24,     16;
  3:   14,    84,    144,      64;
  4:   34,   272,    736,     768,     256;
  5:   82,   820,   3040,    5120,    3840,    1024;
  6:  198,  2376,  11184,   26112,   31488,   18432,    4096;
  7:  478,  6692,  38192,  114240,  191744,  179200,   86016,   16384;
  8: 1154, 18464, 123840,  452096,  976384, 1269760,  966656,  393216,   65536;
  9: 2786, 50148, 386496, 1666560, 4409856, 7391232, 7815168, 5013504, 1769472, 262144;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n-2+(4*n-8)*x)*P(n-2,x) + (2*n-4+(4*n-8)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 2 + 4*x, P(2,x) = 6 + 24*x + 16*x^2.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[2, x_] := 6 + 24*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((n - 2 + (4*n - 8)*x)*P[n - 2, x] + (2*n - 4 + (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] = 2 + 4*x; v[3] = 6 + 24*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((n - 2 + (4*n - 8)*x)*v[i - 2] + (2*n - 4 + (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,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