A267482 → D

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

Recurrence

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

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

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

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	1	2													3
3	-2	2	3												3
4	-2	-6	3	4											-1
5	3	-6	-12	4	5										-6
6	3	12	-12	-20	5	6									-6
7	-4	12	30	-20	-30	6	7								1
8	-4	-20	30	60	-30	-42	7	8							9
9	5	-20	-60	60	105	-42	-56	8	9						9
10	5	30	-60	-140	105	168	-56	-72	9	10					-1
11	-6	30	105	-140	-280	168	252	-72	-90	10	11				-12
12	-6	-42	105	280	-280	-504	252	360	-90	-110	11	12			-12
13	7	-42	-168	280	630	-504	-840	360	495	-110	-132	12	13		1
14	7	56	-168	-504	630	1260	-840	-1320	495	660	-132	-156	13	14	15

Generated b-files: A165202 (row sums) ?

Computed in your browser ?

Technical details

Node id: n-bf7c363b1b031b60 · revision 1 · JSON
Definition source: derived:derivative
Fingerprint: f6-eb9d387816209818

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 ?

A165202: 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:  0;
  1:  1;
  2:  1,   2;
  3: -2,   2,   3;
  4: -2,  -6,   3,   4;
  5:  3,  -6, -12,   4,   5;
  6:  3,  12, -12, -20,   5,   6;
  7: -4,  12,  30, -20, -30,   6,   7;
  8: -4, -20,  30,  60, -30, -42,   7, 8;
  9:  5, -20, -60,  60, 105, -42, -56, 8, 9;

%F Formula

The row polynomials satisfy (2*n^2-3*n+1)*P(n,x) = (n+1+(n^2+n)*x)*P(n-1,x) - (4*n-n*x^2)*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[((n + 1 + (n^2 + n)*x)*P[n - 1, x] - (4*n - n*x^2)*D[P[n - 1, x], x])/(2*n^2 - 3*n + 1)]];
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] = ((n + 1 + (n^2 + n)*x)*v[i - 1] - (4*n - n*x^2)*deriv(v[i - 1], x))/(2*n^2 - 3*n + 1)); 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-670b46fe93dc1660
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-670b46fe93dc1660

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-670b46fe93dc1660

Laguerre’s theorem on differential operators [SP14] polytool ?
- premise (sequence): Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where…
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
  - theorem_check three_term_recurrence e-d1dad2b285ccdcec root: theorem check
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ?
  - theorem_check three_term_recurrence e-d92e4a9b9719ac57 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