A116414 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	8	2						10
3	42	24	3					69
4	184	174	48	4				410
5	731	992	444	80	5			2252
6	2736	4908	3120	900	120	6		11790
7	9844	22128	18510	7520	1590	168	7	59767

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-e108138472a4dc34 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-f978c93cbf2e367f

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:      1;
  2:      8,      2;
  3:     42,     24,      3;
  4:    184,    174,     48,      4;
  5:    731,    992,    444,     80,      5;
  6:   2736,   4908,   3120,    900,    120,     6;
  7:   9844,  22128,  18510,   7520,   1590,   168,    7;
  8:  34448,  93476,  97872,  52060,  15400,  2562,  224,   8;
  9: 118101, 376416, 476172, 316176, 121905, 28224, 3864, 288, 9;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 8 + (n + 2)*x)*P[n - 1, x] + (4 + 8*x + x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 8 + (n + 2)*x)*v[i - 1] + (4 + 8*x + x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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-b01f215f4678a533
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-b01f215f4678a533
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-b01f215f4678a533
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-b01f215f4678a533

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-b01f215f4678a533

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check three_term_recurrence e-894010de7c322dc8 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-7bedda0241d23786 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