A053122 → D+1

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

Recurrence

\(\left(12 - 14\,x + 2\,x^{2} - 8\,n + 14\,n x - 2\,n x^{2} - 6\,n^{2} + 2\,n^{3}\right)\,P_{n}(x) = \left(14\,n - 17\,n x + 8\,n x^{2} - n x^{3} - 2\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(8\,x - 22\,x^{2} + 9\,x^{3} - x^{4} + 4\,n x - n x^{2} - 4\,n^{2} x + n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	-1	1							0
3	-1	-2	1						-2
4	6	-2	-3	1					2
5	-15	22	-3	-4	1				1
6	29	-77	52	-4	-5	1			-4
7	-49	196	-234	100	-5	-6	1		3
8	76	-420	738	-550	170	-6	-7	1	2

Show all 15 stored rows

Generated b-files: A122918 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-2eff9cba2593f9a5 · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-7e39731cad3c3538

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 -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 ?

A122918: 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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:    1;
   2:   -1,     1;
   3:   -1,    -2,     1;
   4:    6,    -2,    -3,     1;
   5:  -15,    22,    -3,    -4,     1;
   6:   29,   -77,    52,    -4,    -5,     1;
   7:  -49,   196,  -234,   100,    -5,    -6,   1;
   8:   76,  -420,   738,  -550,   170,    -6,  -7,  1;
   9: -111,   804, -1914,  2068, -1105,   266,  -7, -8,  1;
  10:  155, -1419,  4356, -6292,  4823, -1995, 392, -8, -9, 1;

%F Formula

The row polynomials satisfy (2*n^3-6*n^2-8*n+12+(14*n-14)*x-(2*n-2)*x^2)*P(n,x) = -(2*n^3-14*n-(n^3-17*n)*x-8*n*x^2+n*x^3)*P(n-1,x) - ((4*n^2-4*n-8)*x-(n^2-n-22)*x^2-9*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 14*n - (n^3 - 17*n)*x - 8*n*x^2 + n*x^3)*P[n - 1, x] - ((4*n^2 - 4*n - 8)*x - (n^2 - n - 22)*x^2 - 9*x^3 + x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 - 8*n + 12 + (14*n - 14)*x - (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 14*n - (n^3 - 17*n)*x - 8*n*x^2 + n*x^3)*v[i - 1] - ((4*n^2 - 4*n - 8)*x - (n^2 - n - 22)*x^2 - 9*x^3 + x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 - 8*n + 12 + (14*n - 14)*x - (2*n - 2)*x^2)); 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-93428a39d30e4b96
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-93428a39d30e4b96

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-93428a39d30e4b96

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-46262d3eff45ae1d root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-8de75f6dc51d73d9 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