A358735 → D

nonnegative coefficients

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

Recurrence

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(n - n^{2}\right)\,P_{n-4}(x) + \left(-6\,n + 4\,n x + 4\,n^{2} - 6\,n^{2} x + 2\,n^{3} x\right)\,P_{n-3}(x) + \left(-2 + 12\,n - 8\,n x + 2\,n x^{2} - 6\,n^{2} + 12\,n^{2} x - 5\,n^{2} x^{2} - 4\,n^{3} x + 4\,n^{3} x^{2} - n^{4} x^{2}\right)\,P_{n-2}(x) + \left(4 - 10\,n + 4\,n x + 4\,n^{2} - 6\,n^{2} x + 2\,n^{3} x\right)\,P_{n-1}(x)\)

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

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

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

\(P_{3}(x) = 10 + 32\,x + 18\,x^{2}\)

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	4	4													8
3	10	32	18												60
4	20	140	228	96											484
5	35	448	1488	1712	600										4283
6	56	1176	6780	15232	14040	4320									41604
7	84	2688	24420	92032	160760	126576	35280								441840
8	120	5544	74316	429312	1227800	1792128	1251936	322560							5103716
9	165	10560	199056	1658384	7124600	16717920	21285936	13522176	3265920						63784717
10	220	18876	482196	5541536	33751200	117138912	236362336	269989632	158656320	36288000					858229228
11	286	32032	1076790	16496896	136573920	663075744	1954721440	3497869312	3655168416	2011708800	439084800				12375808436
12	364	52052	2247960	44680896	486850080	3175677024	12956140960	33533357312	54372916224	52722708480	27432725760	5748019200			190475376312
13	455	81536	4434144	111829536	1563190800	13283139552	72121995344	256060082432	595516014720	888820275840	808361097984	400541967360	80951270400		3117335380103
14	560	123760	8332584	261748864	4596548400	49652207520	348245795744	1632538733056	5169605411520	10988500294400	15277121718144	13141214502912	6236789045760	1220496076800	54069030540024

Technical details

Node id: n-8d1e3c59c8ce3c0e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6d8072be47539c36

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:   1;
  2:   4,     4;
  3:  10,    32,     18;
  4:  20,   140,    228,      96;
  5:  35,   448,   1488,    1712,     600;
  6:  56,  1176,   6780,   15232,   14040,     4320;
  7:  84,  2688,  24420,   92032,  160760,   126576,    35280;
  8: 120,  5544,  74316,  429312, 1227800,  1792128,  1251936,   322560;
  9: 165, 10560, 199056, 1658384, 7124600, 16717920, 21285936, 13522176, 3265920;

%F Formula

The row polynomials satisfy (n^2-3*n+2)*P(n,x) = -(n^2-n)*P(n-4,x) + (4*n^2-6*n+(2*n^3-6*n^2+4*n)*x)*P(n-3,x) - (6*n^2-12*n+2+(4*n^3-12*n^2+8*n)*x+(n^4-4*n^3+5*n^2-2*n)*x^2)*P(n-2,x) + (4*n^2-10*n+4+(2*n^3-6*n^2+4*n)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 4 + 4*x, P(3,x) = 10 + 32*x + 18*x^2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 4 + 4*x;
P[3, x_] := 10 + 32*x + 18*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - n)*P[n - 4, x] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 3, x] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*P[n - 2, x] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*P[n - 1, x])/(n^2 - 3*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; v[3] = 4 + 4*x; v[4] = 10 + 32*x + 18*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(n^2 - n)*v[i - 4] + (4*n^2 - 6*n + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 3] - (6*n^2 - 12*n + 2 + (4*n^3 - 12*n^2 + 8*n)*x + (n^4 - 4*n^3 + 5*n^2 - 2*n)*x^2)*v[i - 2] + (4*n^2 - 10*n + 4 + (2*n^3 - 6*n^2 + 4*n)*x)*v[i - 1])/(n^2 - 3*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 (−∞, 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 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m…?
  - 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