A124038 → D

Proved for all n: all roots real · interlacing (weak) what this means

Recurrence

\(\left(-14 + 2\,x + 24\,n - 6\,n x - 10\,n^{2} + 4\,n^{2} x\right)\,P_{n}(x) = \left(4 + 2\,x + 4\,n - 3\,n x + 2\,n x^{2} - 5\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-12 + 3\,x^{2} + 20\,n - 8\,n x - 5\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	-2	2													0
3	-2	-4	3												-3
4	4	-6	-6	4											-4
5	3	12	-12	-8	5										0
6	-6	12	24	-20	-10	6									6
7	-4	-24	30	40	-30	-12	7								7
8	8	-20	-60	60	60	-42	-14	8							0
9	5	40	-60	-120	105	84	-56	-16	9						-9
10	-10	30	120	-140	-210	168	112	-72	-18	10					-10
11	-6	-60	105	280	-280	-336	252	144	-90	-20	11				0
12	12	-42	-210	280	560	-504	-504	360	180	-110	-22	12			12
13	7	84	-168	-560	630	1008	-840	-720	495	220	-132	-24	13		13
14	-14	56	336	-504	-1260	1260	1680	-1320	-990	660	264	-156	-26	14	0

Generated b-files: A110665 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-5515c868dfdae3d4 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-726fd9907bf71db1

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): no
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

A110665: row sums, 15 terms agree (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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: -2,   2;
  3: -2,  -4,   3;
  4:  4,  -6,  -6,    4;
  5:  3,  12, -12,   -8,   5;
  6: -6,  12,  24,  -20, -10,   6;
  7: -4, -24,  30,   40, -30, -12,   7;
  8:  8, -20, -60,   60,  60, -42, -14,   8;
  9:  5,  40, -60, -120, 105,  84, -56, -16, 9;

%F Formula

The row polynomials satisfy -(10*n^2-24*n+14-(4*n^2-6*n+2)*x)*P(n,x) = (4*n+4-(5*n^2+3*n-2)*x+(2*n^2+2*n)*x^2)*P(n-1,x) + (20*n-12-8*n*x-(5*n-3)*x^2+2*n*x^3)*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 + 4 - (5*n^2 + 3*n - 2)*x + (2*n^2 + 2*n)*x^2)*P[n - 1, x] + (20*n - 12 - 8*n*x - (5*n - 3)*x^2 + 2*n*x^3)*D[P[n - 1, x], x])/(-10*n^2 + 24*n - 14 + (4*n^2 - 6*n + 2)*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 4 - (5*n^2 + 3*n - 2)*x + (2*n^2 + 2*n)*x^2)*v[i - 1] + (20*n - 12 - 8*n*x - (5*n - 3)*x^2 + 2*n*x^3)*deriv(v[i - 1], x))/(-10*n^2 + 24*n - 14 + (4*n^2 - 6*n + 2)*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, +inf)
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-66ea761efe9838f4
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-66ea761efe9838f4

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-66ea761efe9838f4

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-66ab36c44d35dd53 root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-cf11db18e382a69a 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