A136394 → D

nonnegative coefficients

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

Recurrence

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

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

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

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

\(P_{3}(x) = 5\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1								0
2	1							1
3	5							5
4	20	6						26
5	84	70						154
6	409	590	45					1044
7	2365	4718	945					8028
8	16064	39340	13440	420				69264
9	125664	354156	170016	13860				663696
10	1112073	3476652	2110290	296100	4725			6999840
11	10976173	37214892	26825370	5385380	225225			80627040
12	119481284	432801138	356056008	92318380	6722100	62370		1007441280
13	1421542628	5443264242	4976691720	1564720300	165465300	4054050		13575738240
14	18348340113	73685797978	73484578167	26877582732	3729300575	160810650	945945	196287356160

Technical details

Node id: n-d68e6fcf9b7ad391 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-7a2ba97e1e8344f1

Observed on the stored rows ?

Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
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:      0;
  2:      1;
  3:      5;
  4:     20,      6;
  5:     84,     70;
  6:    409,    590,     45;
  7:   2365,   4718,    945;
  8:  16064,  39340,  13440,   420;
  9: 125664, 354156, 170016, 13860;

%F Formula

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

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 1;
P[3, x_] := 5;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*P[n - 4, x] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*P[n - 3, x] - (n^2 - 1 - (2*n - 2)*x)*P[n - 2, x] + (2*n - 1)*P[n - 1, x]];
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] = 0; v[3] = 1; v[4] = 5; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 4*n + 3 - (2*n^2 - 8*n + 6)*x + (n^2 - 4*n + 3)*x^2)*v[i - 4] + (2*n^2 - 6*n + 4 - (2*n^2 - 6*n + 4)*x)*v[i - 3] - (n^2 - 1 - (2*n - 2)*x)*v[i - 2] + (2*n - 1)*v[i - 1]); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[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.