A039991 → D+1

Proved for all n: all roots real what this means

Recurrence

\(\left(-3 - x + 4\,x^{2} - x^{3} + 5\,n + 4\,n x - 6\,n x^{2} + n x^{3} - 2\,n^{2} - 4\,n^{2} x + 2\,n^{2} x^{2} + n^{3} x\right)\,P_{n}(x) = \left(-6 + 2\,x + 9\,x^{2} - 3\,x^{3} - 2\,x^{4} + x^{5} + 10\,n + n x - 15\,n x^{2} + n x^{3} + 4\,n x^{4} - n x^{5} - 4\,n^{2} - 5\,n^{2} x + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} - 2\,n^{2} x^{4} + 2\,n^{3} x - n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x + x^{2} + 3\,x^{3} - 2\,x^{4} - x^{5} + x^{6} + 2\,n x + n x^{2} - 4\,n x^{3} - n x^{4} + 2\,n x^{5} - n^{2} x^{2} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0	1							1
1	1							1
2	2	-2	-1					-1
3	4	-6	-3					-5
4	8	-16	-8	4	1			-11
5	16	-40	-20	20	5			-19
6	32	-96	-48	72	18	-6	-1	-29
7	64	-224	-112	224	56	-42	-7	-41

Show all 15 stored rows

Generated b-files: A110331 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-0b1520d85356534b · revision 1 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-22b73645364b2fe7

Observed on the stored rows ?

Degrees: 0, 0, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12, 14 (irregular)
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 ?

A110331: row sums, 14 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:   1;
  1:   1;
  2:   2,    -2,   -1;
  3:   4,    -6,   -3;
  4:   8,   -16,   -8,    4,   1;
  5:  16,   -40,  -20,   20,   5;
  6:  32,   -96,  -48,   72,  18,   -6,   -1;
  7:  64,  -224, -112,  224,  56,  -42,   -7;
  8: 128,  -512, -256,  640, 160, -192,  -32,  8, 1;
  9: 256, -1152, -576, 1728, 432, -720, -120, 72, 9;

%F Formula

The row polynomials satisfy -(2*n^2-5*n+3-(n^3-4*n^2+4*n-1)*x-(2*n^2-6*n+4)*x^2-(n-1)*x^3)*P(n,x) = -(4*n^2-10*n+6-(2*n^3-5*n^2+n+2)*x-(6*n^2-15*n+9)*x^2+(n^3-3*n^2-n+3)*x^3+(2*n^2-4*n+2)*x^4+(n-1)*x^5)*P(n-1,x) + ((2*n-2)*x-(n^2-n-1)*x^2-(4*n-3)*x^3+(n^2-n-2)*x^4+(2*n-1)*x^5+x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 10*n + 6 - (2*n^3 - 5*n^2 + n + 2)*x - (6*n^2 - 15*n + 9)*x^2 + (n^3 - 3*n^2 - n + 3)*x^3 + (2*n^2 - 4*n + 2)*x^4 + (n - 1)*x^5)*P[n - 1, x] + ((2*n - 2)*x - (n^2 - n - 1)*x^2 - (4*n - 3)*x^3 + (n^2 - n - 2)*x^4 + (2*n - 1)*x^5 + x^6)*D[P[n - 1, x], x])/(-2*n^2 + 5*n - 3 + (n^3 - 4*n^2 + 4*n - 1)*x + (2*n^2 - 6*n + 4)*x^2 + (n - 1)*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; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^2 - 10*n + 6 - (2*n^3 - 5*n^2 + n + 2)*x - (6*n^2 - 15*n + 9)*x^2 + (n^3 - 3*n^2 - n + 3)*x^3 + (2*n^2 - 4*n + 2)*x^4 + (n - 1)*x^5)*v[i - 1] + ((2*n - 2)*x - (n^2 - n - 1)*x^2 - (4*n - 3)*x^3 + (n^2 - n - 2)*x^4 + (2*n - 1)*x^5 + x^6)*deriv(v[i - 1], x))/(-2*n^2 + 5*n - 3 + (n^3 - 4*n^2 + 4*n - 1)*x + (2*n^2 - 6*n + 4)*x^2 + (n - 1)*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-77a19b2535dc4780

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-77a19b2535dc4780

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of cos(x)^n in polynomial for cos(nx).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-9aecc6cc430e81f1 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