A053122 → x ↦ x+1 → D

Proved for all n all roots real interlacing (weak)

Recurrence

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

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

\(P_{2}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	1														1
3	-2	2													0
4	1	-6	3												-2
5	2	6	-12	4											0
6	-4	4	18	-20	5										3
7	2	-18	0	40	-30	6									0
8	3	18	-45	-20	75	-42	7								-4
9	-6	6	72	-80	-70	126	-56	8							0
10	3	-36	-18	196	-105	-168	196	-72	9						5
11	4	36	-108	-140	420	-84	-336	288	-90	10					0
12	-8	8	180	-200	-490	756	42	-600	405	-110	11				-6
13	4	-60	-60	580	-180	-1260	1176	360	-990	550	-132	12			0
14	5	60	-210	-460	1395	288	-2688	1584	990	-1540	726	-156	13		7
15	-10	10	360	-400	-1790	2646	1848	-5016	1782	2090	-2288	936	-182	14	0

Technical details

Node id: n-faea90cc70c688ad · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-2e9c53d04715c401

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 -2)
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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:  0;
   2:  1;
   3: -2,   2;
   4:  1,  -6,   3;
   5:  2,   6, -12,   4;
   6: -4,   4,  18, -20,    5;
   7:  2, -18,   0,  40,  -30,    6;
   8:  3,  18, -45, -20,   75,  -42,   7;
   9: -6,   6,  72, -80,  -70,  126, -56,   8;
  10:  3, -36, -18, 196, -105, -168, 196, -72, 9;

%F Formula

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

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 - (n + 1)*x)*P[n - 1, x] - (3 + 2*x - x^2)*D[P[n - 1, x], x])/(2*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(n + 1 - (n + 1)*x)*v[i - 1] - (3 + 2*x - x^2)*deriv(v[i - 1], x))/(2*n - 4)); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, +∞)
by Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from shift_x_plus_one of Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increa…?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).?
    
    Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check

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