login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

%I #17 Nov 10 2013 01:32:32

%S 2,2,18,2,123,52,2,843,724,110,2,5778,10084,2525,198,2,39603,140452,

%T 57965,6726,322,2,271443,1956244,1330670,228486,15127,488,2,1860498,

%U 27246964,30547445,7761798,710647,30248,702,2,12752043,379501252,701260565,263672646

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

%H Ralf Stephan, <a href="/A231123/a231123.pdf">On a class of reducible trinomials</a>

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

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

%e Array starts

%e 2, 18, 52, 110, 198, 322, 488, 702, 970,...

%e 2, 123, 724, 2525, 6726, 15127, 30248, 55449, 95050,...

%e 2, 843, 10084, 57965, 228486, 710647, 1874888, 4379769, 9313930,...

%e 2, 5778, 140452, 1330670, 7761798, 33385282, 116212808, 345946302,...

%e 2, 39603, 1956244, 30547445, 263672646, 1568397607, 7203319208,...

%o (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 nonn,tabl

%O 2,1

%A _Ralf Stephan_, Nov 04 2013

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)