A165253 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−6, −2)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(-5 - x + 4\,n + n x\right)\,P_{n-1}(x) + \left(12 + 8\,x + x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	3	1						4
3	11	7	1					19
4	41	38	11	1				91
5	153	186	81	15	1			436
6	571	859	499	140	19	1		2089
7	2131	3821	2774	1044	215	23	1	10009

All 15 rows

Technical details

Node id: n-8e2b6cb40f5d380a · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d02bc8cc60e9d4df

Observed on the stored rows ?

Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
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:     1;
  2:     3,     1;
  3:    11,     7,     1;
  4:    41,    38,    11,     1;
  5:   153,   186,    81,    15,     1;
  6:   571,   859,   499,   140,    19,    1;
  7:  2131,  3821,  2774,  1044,   215,   23,   1;
  8:  7953, 16556, 14418,  6810,  1885,  306,  27,  1;
  9: 29681, 70356, 71454, 40614, 14135, 3086, 413, 31, 1;

%F Formula

The row polynomials satisfy (2*n-3)*P(n,x) = (4*n-5+(n-1)*x)*P(n-1,x) + (12+8*x+x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 5 + (n - 1)*x)*P[n - 1, x] + (12 + 8*x + x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n - 5 + (n - 1)*x)*v[i - 1] + (12 + 8*x + x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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 (-6, -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)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (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)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from shift_x_plus_one of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where …?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operato…?
    
    ↑ Three-term recurrence criterion (above)

References

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