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	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

All 15 rows

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