A078621 Values of n such that Sum[ -(-1)^(k) n/k (n-1)/(k+1),{k,1,n}] (n!!) is an integer.

1, 5, 9, 11, 13, 17, 25, 33, 49, 65, 97, 129, 193, 257, 385, 513, 769, 1025, 1537, 2049, 3073, 4097, 6145, 8193, 12289, 16385, 24577, 32769, 49153, 65537, 98305, 131073, 196609, 262145, 393217, 524289, 786433, 1048577, 1572865, 2097153, 3145729

Offset

1,2

Comments

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))

Links

Table of n, a(n) for n=1..41.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Formula

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)

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

Mathematica

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]

Crossrefs

Sequence in context: A161155 A337920 A314585 * A287521 A287529 A179383

Adjacent sequences: A078618 A078619 A078620 * A078622 A078623 A078624

Keyword

easy,nonn

Author

Wouter Meeussen, Dec 11 2002

Status

approved