cp's OEIS Frontend

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.

%I A231123 #17 Nov 10 2013 01:32:32
%S A231123 2,2,18,2,123,52,2,843,724,110,2,5778,10084,2525,198,2,39603,140452,
%T A231123 57965,6726,322,2,271443,1956244,1330670,228486,15127,488,2,1860498,
%U A231123 27246964,30547445,7761798,710647,30248,702,2,12752043,379501252,701260565,263672646
%N 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.
%C A231123 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.
%D A231123 A. Schinzel, On reducible trinomials III. In: Selecta, Vol. I, European Mathematical Society 2007, pp. 625-626.
%H A231123 Ralf Stephan, <a href="/A231123/a231123.pdf">On a class of reducible trinomials</a>
%F A231123 T(,2) = 2, T(1,n) = A121670(n), T(2,n) = A230586(n).
%F A231123 T(n,k) = sum(i=1..n, (-1)^i * A111125(n,i) * k^(2i+1) ).
%e A231123 Array starts
%e A231123 2, 18, 52, 110, 198, 322, 488, 702, 970,...
%e A231123 2, 123, 724, 2525, 6726, 15127, 30248, 55449, 95050,...
%e A231123 2, 843, 10084, 57965, 228486, 710647, 1874888, 4379769, 9313930,...
%e A231123 2, 5778, 140452, 1330670, 7761798, 33385282, 116212808, 345946302,...
%e A231123 2, 39603, 1956244, 30547445, 263672646, 1568397607, 7203319208,...
%o A231123 (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))
%K A231123 nonn,tabl
%O A231123 2,1
%A A231123 _Ralf Stephan_, Nov 04 2013