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

2,1

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

Table of n, a(n) for n=2..42.

Ralf Stephan, On a class of reducible trinomials

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) ).

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

Sequence in context: A074970 A297794 A291765 * A225123 A087338 A055735

Adjacent sequences: A231120 A231121 A231122 * A231124 A231125 A231126

Keyword

nonn,tabl

Author

Ralf Stephan, Nov 04 2013

Status

approved