A207607 → D

nonnegative coefficients

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence

\(\left(2 - 4\,x + 3\,n + 12\,n x - 3\,n^{2} - 12\,n^{2} x - 2\,n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(-3\,n - 3\,n x - 7\,n^{2} - 12\,n^{2} x - 2\,n^{2} x^{2} - 2\,n^{3} + 3\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-12\,x - 3\,x^{2} - 8\,n x - 10\,n x^{2} - 2\,n x^{3} - 4\,n^{2} x + 7\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	2							2
2	5	4						9
3	9	18	6					33
4	14	50	39	8				111
5	20	110	147	68	10			355
6	27	210	420	324	105	12		1098
7	35	364	1008	1140	605	150	14	3316

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-2ddf2e5140d53d69 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-92685669426d3e72

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 -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

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  0;
  1:  2;
  2:  5,   4;
  3:  9,  18,    6;
  4: 14,  50,   39,    8;
  5: 20, 110,  147,   68,   10;
  6: 27, 210,  420,  324,  105,   12;
  7: 35, 364, 1008, 1140,  605,  150,   14;
  8: 44, 588, 2142, 3300, 2530, 1014,  203,  16;
  9: 54, 900, 4158, 8316, 8580, 4914, 1575, 264, 18;

%F Formula

The row polynomials satisfy -(2*n^3+3*n^2-3*n-2-(4*n^3-12*n^2+12*n-4)*x)*P(n,x) = -(2*n^3+7*n^2+3*n-(3*n^3-12*n^2-3*n)*x-(2*n^3-2*n^2)*x^2)*P(n-1,x) - ((4*n^2+8*n+12)*x-(7*n^2-10*n-3)*x^2-(2*n^2-2*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 7*n^2 + 3*n - (3*n^3 - 12*n^2 - 3*n)*x - (2*n^3 - 2*n^2)*x^2)*P[n - 1, x] - ((4*n^2 + 8*n + 12)*x - (7*n^2 - 10*n - 3)*x^2 - (2*n^2 - 2*n)*x^3)*D[P[n - 1, x], x])/(-2*n^3 - 3*n^2 + 3*n + 2 + (4*n^3 - 12*n^2 + 12*n - 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 + 7*n^2 + 3*n - (3*n^3 - 12*n^2 - 3*n)*x - (2*n^3 - 2*n^2)*x^2)*v[i - 1] - ((4*n^2 + 8*n + 12)*x - (7*n^2 - 10*n - 3)*x^2 - (2*n^2 - 2*n)*x^3)*deriv(v[i - 1], x))/(-2*n^3 - 3*n^2 + 3*n + 2 + (4*n^3 - 12*n^2 + 12*n - 4)*x)); 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)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-724418f778f34057
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-724418f778f34057
Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.

derived from the cited steps by a stated theorem: e-724418f778f34057
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-724418f778f34057

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image derivative_polynomial, shift 0 e-724418f778f34057

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-8f881604bf5a9d75 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-3cffc405672d7710 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?

Related sequences ?

No candidate equalities or related sequences.

References

[Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
[Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023