OFFSET
2,1
COMMENTS
The polynomial x^(4n+2) - T(n,k)*x^(2n+1) + 1 is reducible. Example: x^10-123x^5+1=(x^2-3x+1)(x^8+3x^7+8x^6+21x^5+55x^4+21x^3+8x^2+3x+1). It is conjectured that for prime p=2n+1, these are the only values where this holds.
REFERENCES
A. Schinzel, On reducible trinomials III. In: Selecta, Vol. I, European Mathematical Society 2007, pp. 625-626.
LINKS
FORMULA
EXAMPLE
Array starts
2, 18, 52, 110, 198, 322, 488, 702, 970,...
2, 123, 724, 2525, 6726, 15127, 30248, 55449, 95050,...
2, 843, 10084, 57965, 228486, 710647, 1874888, 4379769, 9313930,...
2, 5778, 140452, 1330670, 7761798, 33385282, 116212808, 345946302,...
2, 39603, 1956244, 30547445, 263672646, 1568397607, 7203319208,...
PROG
(PARI) T(i, k)=n=2*i+1; sum(m=0, (n-1)/2, (-1)^(m+(n-1)/2)*n*binomial((n+2*m+1)/2-1, 2*m)/(2*m+1)*k^(2*m+1))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, Nov 04 2013
STATUS
approved