A179900 → D

Proved for all n all roots real interlacing (weak)

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(10 - 2\,x + 5\,n - n x\right)\,P_{n-1}(x) + \left(-21 + 10\,x - x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	-1							-1
2	-10	2						-8
3	-73	30	-3					-46
4	-470	294	-60	4				-232
5	-2828	2380	-738	100	-5			-1091
6	-16310	17262	-7200	1480	-150	6		-4912
7	-91371	116550	-61155	16900	-2595	210	-7	-21468

All 15 rows

Technical details

Node id: n-da081fb7592ee6c9 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-e589bedfdbabf881

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        0;
  1:       -1;
  2:      -10,       2;
  3:      -73,      30,       -3;
  4:     -470,     294,      -60,       4;
  5:    -2828,    2380,     -738,     100,      -5;
  6:   -16310,   17262,    -7200,    1480,    -150,     6;
  7:   -91371,  116550,   -61155,   16900,   -2595,   210,    -7;
  8:  -501150,  748230,  -473400,  164560,  -33950,  4158,  -280,   8;
  9: -2704755, 4626900, -3428190, 1437100, -372855, 61320, -6244, 360, -9;

%F Formula

The row polynomials satisfy (2*n-2)*P(n,x) = (5*n+10-(n+2)*x)*P(n-1,x) - (21-10*x+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[((5*n + 10 - (n + 2)*x)*P[n - 1, x] - (21 - 10*x + x^2)*D[P[n - 1, x], x])/(2*n - 2)]];
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] = ((5*n + 10 - (n + 2)*x)*v[i - 1] - (21 - 10*x + x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, 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 ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - 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.
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[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