A124038 → D+1

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

Recurrence

\(\left(24 + 16\,x - 12\,x^{2} + 2\,x^{3} - 38\,n - 12\,n x + 18\,n x^{2} - 4\,n x^{3} + 24\,n^{2} - 6\,n^{2} x - 10\,n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(12 + 8\,x - 6\,x^{2} + x^{3} - 16\,n - 2\,n x - 2\,n x^{2} + 7\,n x^{3} - 2\,n x^{4} + 4\,n^{2} - 3\,n^{2} x + 2\,n^{2} x^{2} - 5\,n^{3} x + 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(48 - 12\,x - 32\,x^{2} + 11\,x^{3} + 5\,x^{4} - 2\,x^{5} - 12\,n + 3\,n x^{2} + 20\,n^{2} - 8\,n^{2} x - 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-1	1							0
2	-3	0	1						-2
3	0	-6	1	1					-4
4	5	-2	-9	2	1				-3
5	1	15	-6	-12	3	1			2
6	-7	6	30	-12	-15	4	1		7
7	-2	-28	18	50	-20	-18	5	1	6

Show all 15 stored rows

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Technical details

Node id: n-3b88797d535e9a6e · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-f9962005bbbd3301

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  1;
  1: -1,   1;
  2: -3,   0,   1;
  3:  0,  -6,   1,    1;
  4:  5,  -2,  -9,    2,   1;
  5:  1,  15,  -6,  -12,   3,   1;
  6: -7,   6,  30,  -12, -15,   4,   1;
  7: -2, -28,  18,   50, -20, -18,   5,   1;
  8:  9, -12, -70,   40,  75, -30, -21,   6, 1;
  9:  3,  45, -40, -140,  75, 105, -42, -24, 7, 1;

%F Formula

The row polynomials satisfy -(10*n^3-24*n^2+38*n-24-(4*n^3-6*n^2-12*n+16)*x-(18*n-12)*x^2+(4*n-2)*x^3)*P(n,x) = (4*n^2-16*n+12-(5*n^3+3*n^2+2*n-8)*x+(2*n^3+2*n^2-2*n-6)*x^2+(7*n+1)*x^3-2*n*x^4)*P(n-1,x) + (20*n^2-12*n+48-(8*n^2+12)*x-(5*n^2-3*n+32)*x^2+(2*n^2+11)*x^3+5*x^4-2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 16*n + 12 - (5*n^3 + 3*n^2 + 2*n - 8)*x + (2*n^3 + 2*n^2 - 2*n - 6)*x^2 + (7*n + 1)*x^3 - 2*n*x^4)*P[n - 1, x] + (20*n^2 - 12*n + 48 - (8*n^2 + 12)*x - (5*n^2 - 3*n + 32)*x^2 + (2*n^2 + 11)*x^3 + 5*x^4 - 2*x^5)*D[P[n - 1, x], x])/(-10*n^3 + 24*n^2 - 38*n + 24 + (4*n^3 - 6*n^2 - 12*n + 16)*x + (18*n - 12)*x^2 - (4*n - 2)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 - 16*n + 12 - (5*n^3 + 3*n^2 + 2*n - 8)*x + (2*n^3 + 2*n^2 - 2*n - 6)*x^2 + (7*n + 1)*x^3 - 2*n*x^4)*v[i - 1] + (20*n^2 - 12*n + 48 - (8*n^2 + 12)*x - (5*n^2 - 3*n + 32)*x^2 + (2*n^2 + 11)*x^3 + 5*x^4 - 2*x^5)*deriv(v[i - 1], x))/(-10*n^3 + 24*n^2 - 38*n + 24 + (4*n^3 - 6*n^2 - 12*n + 16)*x + (18*n - 12)*x^2 - (4*n - 2)*x^3)); 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-fa558ee223516e16
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-fa558ee223516e16

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-fa558ee223516e16

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

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-1	1							0
2	-3	0	1						-2
3	0	-6	1	1					-4
4	5	-2	-9	2	1				-3
5	1	15	-6	-12	3	1			2
6	-7	6	30	-12	-15	4	1		7
7	-2	-28	18	50	-20	-18	5	1	6

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-1	1							0
2	-3	0	1						-2
3	0	-6	1	1					-4
4	5	-2	-9	2	1				-3
5	1	15	-6	-12	3	1			2
6	-7	6	30	-12	-15	4	1		7
7	-2	-28	18	50	-20	-18	5	1	6

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	-1	1							0
2	-3	0	1						-2
3	0	-6	1	1					-4
4	5	-2	-9	2	1				-3
5	1	15	-6	-12	3	1			2
6	-7	6	30	-12	-15	4	1		7
7	-2	-28	18	50	-20	-18	5	1	6