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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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

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Last modified October 21 21:46 EDT 2019. Contains 328315 sequences. (Running on oeis4.)