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	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	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
8	44	588	2142	3300	2530	1014	203	16							9837
9	54	900	4158	8316	8580	4914	1575	264	18						28779
10	65	1320	7524	18876	25025	19110	8680	2312	333	20					83265
11	77	1870	12870	39468	65065	63336	38080	14280	3249	410	22				238727
12	90	2574	21021	77220	154440	185640	141372	69768	22230	4410	495	24			679284
13	104	3458	33033	143000	340340	493272	461244	286824	119700	33110	5819	588	26		1920518
14	119	4550	50232	252824	704990	1209312	1356600	1031016	539847	194810	47564	7500	689	28	5400081

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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