A077155 - OEIS

A077155

Let p(2n+1,x)=(x+1)(x+2)...(x+2n)(x+2n+1), a(n) is the smallest integer >0 such that p(2n+1,x)-k has only one real root for any k >=a(n).

%I #11 Jan 22 2015 09:49:08

%S 1,4,96,4930,416615,52346851,9150486666,2122773858331,630854176216923,

%T 233667907156182198,105531126177212999940,57078667671269237092154,

%U 36423221938771375213756343,27076505528935399371748578683

%N Let p(2n+1,x)=(x+1)(x+2)...(x+2n)(x+2n+1), a(n) is the smallest integer >0 such that p(2n+1,x)-k has only one real root for any k >=a(n).

%C a(n) is the smallest integer strictly greater than the maximum value of p(2n+1,x) in the interval [ -1,-(2n+1)]. Note that this maximum value is attained by p(2n+1,x) at some root of its derivative. - _Max Alekseyev_, Oct 18 2008

%o (PARI) {a(n) = local(p,r,m); p=prod(k=1,2*n+1,x+k); r=real(polroots(deriv(p))); m=vecmax(vector(#r,j,floor(subst(p,x,r[j])))); if( polsturm(p-m)<=1 || polsturm(p-m-1)>1, error("increase realprecision")); m+1} \\ _Max Alekseyev_, Oct 18 2008

%K nonn

%O 1,2

%A _Benoit Cloitre_, Nov 29 2002

%E a(5)-a(13) from _Max Alekseyev_, Oct 18 2008