A078621 - OEIS

%I #8 Jun 13 2015 00:50:49

%S 1,5,9,11,13,17,25,33,49,65,97,129,193,257,385,513,769,1025,1537,2049,

%T 3073,4097,6145,8193,12289,16385,24577,32769,49153,65537,98305,131073,

%U 196609,262145,393217,524289,786433,1048577,1572865,2097153,3145729

%N Values of n such that Sum[ -(-1)^(k) n/k (n-1)/(k+1),{k,1,n}] (n!!) is an integer.

%C Sum[ -(-1)^(k) n/k(n-1)/(k+1), {k,1,n}] can be simplified to -((-1+n)*n*(1+(2*(-1)^n)/n +(-2+(-1)^n)*Log[2]- ((-1)^n*(-3*PolyGamma[0,n/2]+ 2*PolyGamma[0,n]+ PolyGamma[0,(3+n)/2]))/2))

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2).

%F G.f.: -(4*x^6-6*x^4-6*x^3+2*x^2+4*x+1)/(-2*x^3+2*x^2+x-1)

%F For n>4, a(n) = (4 - n%2) * 2^floor((n-1)/2) + 1. - _Ralf Stephan_, Mar 17 2004

%t Position[Table[Sum[ -(-1)^(k) n/k(n-1)/(k+1), {k, 1, n}] (n!!), {n, 1, 1025}], _Integer]// Flatten or, equivalently, CoefficientList[Series[ -(4x^6-6*x^4-6*x^3+2*x^2+4*x+1)/(-2*x^3+2*x^2+x-1), {x, 0, 48}], x]

%K easy,nonn

%O 1,2

%A _Wouter Meeussen_, Dec 11 2002