A122542 → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 4\,x - 4\,x^{2}\right)\,P_{n-4}(x) + \left(-4 - 12\,x - 8\,x^{2}\right)\,P_{n-3}(x) + \left(-2 - 4\,x - 4\,x^{2}\right)\,P_{n-2}(x) + \left(4 + 4\,x\right)\,P_{n-1}(x)\)

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

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

\(P_{3}(x) = 11 + 16\,x + 4\,x^{2}\)

\(P_{4}(x) = 33 + 82\,x + 52\,x^{2} + 8\,x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	3	2							5
3	11	16	4						31
4	33	82	52	8					175
5	97	336	368	144	16				961
6	275	1226	1936	1312	368	32			5149
7	763	4160	8612	8496	4112	896	64		27103
8	2081	13418	34420	44936	31728	11872	2112	128	140695

All 15 rows

Technical details

Node id: n-9dd9e585db514cc1 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-e553b6ffd2273fac

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 -1)
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:     1;
   2:     3,      2;
   3:    11,     16,      4;
   4:    33,     82,     52,      8;
   5:    97,    336,    368,    144,      16;
   6:   275,   1226,   1936,   1312,     368,     32;
   7:   763,   4160,   8612,   8496,    4112,    896,     64;
   8:  2081,  13418,  34420,  44936,   31728,  11872,   2112,   128;
   9:  5601,  41696, 127680, 208096,  195232, 106496,  32384,  4864,   256;
  10: 14915, 125906, 448256, 877312, 1031776, 748864, 331392, 84736, 11008, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(1+4*x+4*x^2)*P(n-4,x) - (4+12*x+8*x^2)*P(n-3,x) - (2+4*x+4*x^2)*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 3 + 2*x, P(3,x) = 11 + 16*x + 4*x^2, P(4,x) = 33 + 82*x + 52*x^2 + 8*x^3.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[3, x_] := 11 + 16*x + 4*x^2;
P[4, x_] := 33 + 82*x + 52*x^2 + 8*x^3;
P[n_, x_] := P[n, x] = Expand[-(1 + 4*x + 4*x^2)*P[n - 4, x] - (4 + 12*x + 8*x^2)*P[n - 3, x] - (2 + 4*x + 4*x^2)*P[n - 2, x] + (4 + 4*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] = 1; v[2] = 3 + 2*x; v[3] = 11 + 16*x + 4*x^2; v[4] = 33 + 82*x + 52*x^2 + 8*x^3; for(n = 5, N, my(i = n); v[i] = -(1 + 4*x + 4*x^2)*v[i - 4] - (4 + 12*x + 8*x^2)*v[i - 3] - (2 + 4*x + 4*x^2)*v[i - 2] + (4 + 4*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_A013609 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.