A085841 → D

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(3 - 5\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(16\,x + n - 52\,n x + 2\,n^{2} + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 8\,x^{2} + 4\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	4														4
2	40	32													72
3	140	672	192												1004
4	336	4032	6912	1024											12304
5	660	14784	63360	56320	5120										140244
6	1144	41184	329472	732160	399360	24576									1527896
7	1820	96096	1235520	5125120	6988800	2580480	114688								16142524
8	2720	198016	3734016	24893440	63365120	58490880	15597568	524288							166806048
9	3876	372096	9674496	94595072	386979840	666796032	444530688	89653248	2359296						1694964644
10	5320	651168	22325760	300984320	1805905920	5000970240	6223429632	3137863680	495452160	10485760					16998073960
11	7084	1076768	47070144	836802560	6922639360	28116566016	56233132032	52925300736	20891566080	2652897280	46137344				168673195404
12	9200	1700160	92294400	2092006400	22821888000	127802572800	374887546880	567056793600	417831321600	132644864000	13841203200	201326592			1659273526832
13	11700	2583360	170501760	4799308800	66754022400	492952780800	1993720135680	4423042990080	5237814067200	3103889817600	809710387200	70665633792	872415232		16204394655604
14	14616	3800160	299669760	10255365120	177138124800	1667823575040	8895059066880	27208415969280	47256722472960	45006402355200	21916161146880	4781707886592	354200584192	3758096384	157277948127864

Technical details

Node id: n-e8bc601e49e62912 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d3ae4fc63851c32e

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): yes
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:    4;
  2:   40,     32;
  3:  140,    672,     192;
  4:  336,   4032,    6912,     1024;
  5:  660,  14784,   63360,    56320,      5120;
  6: 1144,  41184,  329472,   732160,    399360,     24576;
  7: 1820,  96096, 1235520,  5125120,   6988800,   2580480,    114688;
  8: 2720, 198016, 3734016, 24893440,  63365120,  58490880,  15597568,   524288;
  9: 3876, 372096, 9674496, 94595072, 386979840, 666796032, 444530688, 89653248, 2359296;

%F Formula

The row polynomials satisfy (2*n^2-5*n+3)*P(n,x) = (2*n^2+n+(24*n^2-52*n+16)*x)*P(n-1,x) + ((4*n-2)*x-(16*n-8)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + n + (24*n^2 - 52*n + 16)*x)*P[n - 1, x] + ((4*n - 2)*x - (16*n - 8)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 5*n + 3)]];
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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + n + (24*n^2 - 52*n + 16)*x)*v[i - 1] + ((4*n - 2)*x - (16*n - 8)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 5*n + 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from 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: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).?
  - 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