OFFSET
0,4
COMMENTS
Define polynomials S_0(x)=1, S_1(x)=x, and S_{n+1}(x)=(x+2n(n+1))S_n(x)-n^4*S_{n-1}(x) for n>0. Then S_n(0)=a(n) and S_n(1)=(n!)^2 for all n.
Conjectures: (i) S_n(x) is irreducible over the field of rational numbers for every n=1,2,3,...
(ii) a(n)=S_n(0) is negative if and only if 1<n<58 or n>2177.
(iii) |a(n)|^{1/n}=o(n^2) as n tends to the infinity.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 0..30
Zhi-Wei Sun, A sequence of irreducible polynomials, a message to Number Theory List, Mar 20 2013.
EXAMPLE
a(2)=2*1*2*a(1)-1^4*a(0)=-1,
a(3)=2*2*3*a(2)-2^4*a(1)=-12.
MATHEMATICA
a[0]=1
a[1]=0
a[n_]:=a[n]=2n(n-1)a[n-1]-(n-1)^4*a[n-2]
Table[a[n], {n, 0, 20}]
RecurrenceTable[{a[0]==1, a[1]==0, a[n+1]==2n(n+1)a[n]-n^4 a[n-1]}, a, {n, 20}] (* Harvey P. Dale, Aug 27 2019 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Zhi-Wei Sun, Mar 20 2013
STATUS
approved