A053124 → D

Proved for all n all roots real interlacing (weak)

Recurrence

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

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

\(P_{2}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	4														4
3	-16	32													16
4	40	-192	192												40
5	-80	672	-1536	1024											80
6	140	-1792	6912	-10240	5120										140
7	-224	4032	-23040	56320	-61440	24576									224
8	336	-8064	63360	-225280	399360	-344064	114688								336
9	-480	14784	-152064	732160	-1863680	2580480	-1835008	524288							480
10	660	-25344	329472	-2050048	6988800	-13762560	15597568	-9437184	2359296						660
11	-880	41184	-658944	5125120	-22364160	58490880	-93585408	89653248	-47185920	10485760					880
12	1144	-64064	1235520	-11714560	63365120	-210567168	444530688	-597688320	495452160	-230686720	46137344				1144
13	-1456	96096	-2196480	24893440	-162938880	666796032	-1778122752	3137863680	-3633315840	2652897280	-1107296256	201326592			1456
14	1820	-139776	3734016	-49786880	386979840	-1905131520	6223429632	-13806600192	20891566080	-21223178240	13841203200	-5234491392	872415232		1820
15	-2240	198016	-6110208	94595072	-859955200	5000970240	-19559350272	52925300736	-100279517184	132644864000	-119957094400	70665633792	-24427626496	3758096384	2240

Technical details

Node id: n-8cd2df6bfef06d3c · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-432d9f3d349b8d3a

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:    4;
   3:  -16,     32;
   4:   40,   -192,     192;
   5:  -80,    672,   -1536,     1024;
   6:  140,  -1792,    6912,   -10240,     5120;
   7: -224,   4032,  -23040,    56320,   -61440,     24576;
   8:  336,  -8064,   63360,  -225280,   399360,   -344064,   114688;
   9: -480,  14784, -152064,   732160, -1863680,   2580480, -1835008,   524288;
  10:  660, -25344,  329472, -2050048,  6988800, -13762560, 15597568, -9437184, 2359296;

%F Formula

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

%t Mathematica

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

%o PARI/GP

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

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 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in increasing order).?
  - Three-term recurrence criterion [Sze39, §3.3; Fav35] polytool ? step › direct check
  - 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

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