A122542 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 8\,x - 16\,x^{2}\right)\,P_{n-4}(x) + \left(-4 - 24\,x - 32\,x^{2}\right)\,P_{n-3}(x) + \left(-2 - 8\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 8\,x\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 4\)

\(P_{3}(x) = 16 + 32\,x\)

\(P_{4}(x) = 52 + 224\,x + 192\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	4							4
3	16	32						48
4	52	224	192					468
5	160	1024	1920	1024				4128
6	468	3968	11520	13312	5120			34388
7	1328	13984	54912	99328	81920	24576		276048
8	3684	46304	229056	566272	732160	466944	114688	2159108

All 15 rows

Technical details

Node id: n-9d00fdc908e8a564 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-c7760d28a502c84d

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:     0;
   2:     4;
   3:    16,     32;
   4:    52,    224,     192;
   5:   160,   1024,    1920,     1024;
   6:   468,   3968,   11520,    13312,     5120;
   7:  1328,  13984,   54912,    99328,    81920,    24576;
   8:  3684,  46304,  229056,   566272,   732160,   466944,   114688;
   9: 10048, 146688,  874752,  2746368,  4874240,  4866048,  2523136,   524288;
  10: 27044, 449536, 3136512, 11945984, 27043840, 37109760, 30048256, 13107200, 2359296;

%F Formula

The row polynomials satisfy P(n,x) = -(1+8*x+16*x^2)*P(n-4,x) - (4+24*x+32*x^2)*P(n-3,x) - (2+8*x+16*x^2)*P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 4, P(3,x) = 16 + 32*x, P(4,x) = 52 + 224*x + 192*x^2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 4;
P[3, x_] := 16 + 32*x;
P[4, x_] := 52 + 224*x + 192*x^2;
P[n_, x_] := P[n, x] = Expand[-(1 + 8*x + 16*x^2)*P[n - 4, x] - (4 + 24*x + 32*x^2)*P[n - 3, x] - (2 + 8*x + 16*x^2)*P[n - 2, x] + (4 + 8*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; v[3] = 16 + 32*x; v[4] = 52 + 224*x + 192*x^2; for(n = 5, N, my(i = n); v[i] = -(1 + 8*x + 16*x^2)*v[i - 4] - (4 + 24*x + 32*x^2)*v[i - 3] - (2 + 8*x + 16*x^2)*v[i - 2] + (4 + 8*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...]…?
    
    ↑ 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.