A358735 → basis A007318 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−4, −2)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

\(P_{1}(x) = 3 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	1							4
2	17	12	2						31
3	133	146	52	6					337
4	1313	1980	1102	268	24				4687
5	15623	30179	23072	8720	1628	120			79342
6	217409	514264	502980	260244	75088	11448	720		1582153
7	3462921	9719908	11624456	7676044	3021488	708664	91656	5040	36310177

All 15 rows

Technical details

Node id: n-7d3534761470650a · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-efea276bec11ceea

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:          3,          1;
  2:         17,         12,          2;
  3:        133,        146,         52,          6;
  4:       1313,       1980,       1102,        268,         24;
  5:      15623,      30179,      23072,       8720,       1628,        120;
  6:     217409,     514264,     502980,     260244,      75088,      11448,       720;
  7:    3462921,    9719908,   11624456,    7676044,    3021488,     708664,     91656,     5040;
  8:   62115169,  202147448,  286496492,  230904196,  115720048,   36916408,   7318400,   823968,   40320;
  9: 1238840459, 4592265573, 7537632416, 7188876304, 4389517236, 1779099928, 478524016, 82339920, 8222112, 362880;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (2*n + 2 + n*x)*P[n - 1, x]];
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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (2*n + 2 + n*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 - 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 (-4, -2)
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Affine, reversal and Möbius substitutions (elementary), Geometric basis map (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from basis_map_A007318 of Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
    
    ↑ Three-term recurrence criterion (above)
    
    ↑ Liu–Wang theorem for three-term recurrences (above)
    
    ↑ Three-term recurrence criterion (above)
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
    
    ↑ Three-term recurrence criterion (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from basis_map_A007318 of Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3…?
  - ↑ Affine, reversal and Möbius substitutions (elementary) (above)

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[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
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023