A108299 → D

Proved for all n: all roots real what this means

Recurrence

\(\left(12 - 72\,x - 14\,n + 108\,n x + 4\,n^{2} - 52\,n^{2} x + 8\,n^{3} x\right)\,P_{n}(x) = \left(12 - 30\,x - 48\,x^{2} + 96\,x^{3} - 14\,n + 62\,n x + 52\,n x^{2} - 176\,n x^{3} + 4\,n^{2} - 40\,n^{2} x - 12\,n^{2} x^{2} + 96\,n^{2} x^{3} + 8\,n^{3} x - 16\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(3\,x - 8\,x^{2} - 12\,x^{3} + 32\,x^{4} - 2\,n x + 12\,n x^{2} + 8\,n x^{3} - 48\,n x^{4} - 4\,n^{2} x^{2} + 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	-1														-1
3	-1	-2													-3
4	-1	-4	3												-2
5	-1	-6	6	4											3
6	-1	-8	9	12	-5										7
7	-1	-10	12	24	-15	-6									4
8	-1	-12	15	40	-30	-24	7								-5
9	-1	-14	18	60	-50	-60	28	8							-11
10	-1	-16	21	84	-75	-120	70	40	-9						-6
11	-1	-18	24	112	-105	-210	140	120	-45	-10					7
12	-1	-20	27	144	-140	-336	245	280	-135	-60	11				15
13	-1	-22	30	180	-180	-504	392	560	-315	-210	66	12			8
14	-1	-24	33	220	-225	-720	588	1008	-630	-560	231	84	-13		-9
15	-1	-26	36	264	-275	-990	840	1680	-1134	-1260	616	336	-91	-14	-19

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Technical details

Node id: n-86eb0721424572d8 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-0962ebe4daaf5d6e

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 -2)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): no
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:  0;
   2: -1;
   3: -1,  -2;
   4: -1,  -4,  3;
   5: -1,  -6,  6,  4;
   6: -1,  -8,  9, 12,  -5;
   7: -1, -10, 12, 24, -15,   -6;
   8: -1, -12, 15, 40, -30,  -24,  7;
   9: -1, -14, 18, 60, -50,  -60, 28,  8;
  10: -1, -16, 21, 84, -75, -120, 70, 40, -9;

%F Formula

The row polynomials satisfy (4*n^2-14*n+12+(8*n^3-52*n^2+108*n-72)*x)*P(n,x) = (4*n^2-14*n+12+(8*n^3-40*n^2+62*n-30)*x-(12*n^2-52*n+48)*x^2-(16*n^3-96*n^2+176*n-96)*x^3)*P(n-1,x) - ((2*n-3)*x+(4*n^2-12*n+8)*x^2-(8*n-12)*x^3-(16*n^2-48*n+32)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = -1.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := -1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 14*n + 12 + (8*n^3 - 40*n^2 + 62*n - 30)*x - (12*n^2 - 52*n + 48)*x^2 - (16*n^3 - 96*n^2 + 176*n - 96)*x^3)*P[n - 1, x] - ((2*n - 3)*x + (4*n^2 - 12*n + 8)*x^2 - (8*n - 12)*x^3 - (16*n^2 - 48*n + 32)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12 + (8*n^3 - 52*n^2 + 108*n - 72)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = -1; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 14*n + 12 + (8*n^3 - 40*n^2 + 62*n - 30)*x - (12*n^2 - 52*n + 48)*x^2 - (16*n^3 - 96*n^2 + 176*n - 96)*x^3)*v[i - 1] - ((2*n - 3)*x + (4*n^2 - 12*n + 8)*x^2 - (8*n - 12)*x^3 - (16*n^2 - 48*n + 32)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12 + (8*n^3 - 52*n^2 + 108*n - 72)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-df27c1125e074e5e

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-df27c1125e074e5e

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-7467667c9a732a94 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