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A217703 a(0)=1, a(1)=0, and a(n+1) = 2*n*(n+1)*a(n)-n^4*a(n-1) for n>0. 1
1, 0, -1, -12, -207, -5208, -183105, -8631252, -527065119, -40543768944, -3839804164161, -439319226675420, -59761703074829679, -9535927875005350728, -1764223744981737203073, -374641767646124071723812, -90514221380439108521859135, -24687213546502487871399626208, -7548736406543867794442374424961, -2571770772818360404610536945862316, -970786910104750512664483401420017679 (list; graph; refs; listen; history; text; internal format)
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

Sequence in context: A198529 A151590 A297311 * A245911 A127909 A307691

Adjacent sequences:  A217700 A217701 A217702 * A217704 A217705 A217706

KEYWORD

sign

AUTHOR

Zhi-Wei Sun, Mar 20 2013

STATUS

approved

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Last modified October 21 16:50 EDT 2019. Contains 328302 sequences. (Running on oeis4.)