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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, 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.
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
KEYWORD
nonn,tabl
AUTHOR
Ralf Stephan, Nov 04 2013
STATUS
approved