A231123 Array T(n,k) read by antidiagonals: T(n,k) = sum(i=0...n, (-1)^(n+i) * C(n+i,2i) * n/(2i+1) * k^(2i+1) ), n>0, k>1.

2, 2, 18, 2, 123, 52, 2, 843, 724, 110, 2, 5778, 10084, 2525, 198, 2, 39603, 140452, 57965, 6726, 322, 2, 271443, 1956244, 1330670, 228486, 15127, 488, 2, 1860498, 27246964, 30547445, 7761798, 710647, 30248, 702, 2, 12752043, 379501252, 701260565, 263672646

Offset: 2

Author

Ralf Stephan, Nov 04 2013

Keywords

• nonn
• tabl

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.

Examples

			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,...
		

References

• A. Schinzel, On reducible trinomials III. In: Selecta, Vol. I, European Mathematical Society 2007, pp. 625-626.

Links

• Ralf Stephan, On a class of reducible trinomials

Programs

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))

Formula

T(,2) = 2, T(1,n) = A121670(n), T(2,n) = A230586(n).

T(n,k) = sum(i=1..n, (-1)^i * A111125(n,i) * k^(2i+1) ).