A207606 → D

nonnegative coefficients

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

Recurrence

\(\left(-12 + 6\,x - 14\,n + 10\,n x - 2\,n^{2} + 4\,n^{2} x\right)\,P_{n}(x) = \left(-42 - 3\,x + 12\,x^{2} - 20\,n + 8\,n x + 10\,n x^{2} - 2\,n^{2} + 3\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-20\,x + 11\,x^{2} + 4\,x^{3} - 4\,n x + 7\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 7 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	7	4								11
1	16	22	6							44
2	30	72	45	8						155
3	50	182	192	76	10					510
4	77	392	612	400	115	12				1608
5	112	756	1620	1540	720	162	14			4924
6	156	1344	3762	4840	3250	1176	217	16		14761
7	210	2244	7920	13156	11830	6090	1792	280	18	43540

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-2a27dd49df89ad52 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-54cff7093f65f46e

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:   7,    4;
  1:  16,   22,     6;
  2:  30,   72,    45,     8;
  3:  50,  182,   192,    76,     10;
  4:  77,  392,   612,   400,    115,    12;
  5: 112,  756,  1620,  1540,    720,   162,    14;
  6: 156, 1344,  3762,  4840,   3250,  1176,   217,    16;
  7: 210, 2244,  7920, 13156,  11830,  6090,  1792,   280,   18;
  8: 275, 3564, 15444, 32032,  36855, 25200, 10472,  2592,  351,  20;
  9: 352, 5434, 28314, 71500, 101920, 88536, 48552, 16872, 3600, 430, 22;

%F Formula

The row polynomials satisfy -(2*n^2+14*n+12-(4*n^2+10*n+6)*x)*P(n,x) = -(2*n^2+20*n+42-(3*n^2+8*n-3)*x-(2*n^2+10*n+12)*x^2)*P(n-1,x) - ((4*n+20)*x-(7*n+11)*x^2-(2*n+4)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 7 + 4*x.

%t Mathematica

P[0, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 + 20*n + 42 - (3*n^2 + 8*n - 3)*x - (2*n^2 + 10*n + 12)*x^2)*P[n - 1, x] - ((4*n + 20)*x - (7*n + 11)*x^2 - (2*n + 4)*x^3)*D[P[n - 1, x], x])/(-2*n^2 - 14*n - 12 + (4*n^2 + 10*n + 6)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 7 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^2 + 20*n + 42 - (3*n^2 + 8*n - 3)*x - (2*n^2 + 10*n + 12)*x^2)*v[i - 1] - ((4*n + 20)*x - (7*n + 11)*x^2 - (2*n + 4)*x^3)*deriv(v[i - 1], x))/(-2*n^2 - 14*n - 12 + (4*n^2 + 10*n + 6)*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-6f823d418d8af5c4
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-6f823d418d8af5c4
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-6f823d418d8af5c4
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-6f823d418d8af5c4

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-6f823d418d8af5c4

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the 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-62ac8b7225a515c7 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-3222de37fcca30b3 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