0,2

Related to the formulae sum(k>0,sin(kx)/k^(2n+1))=(-1)^(n+1)/2*x^(2n+1)/(2n+1)!*sum(i=0,2n,(2Pi/x)^i*B(i)*C(2n+1,i)) and if x=Pi/2 sum(k>0,(-1)^(k+1)/k^(2n+1))=(-1)^n*E(2n)*Pi^(2n+1)/2^(2n+2)/(2n)! - Benoit Cloitre (benoit7848c(AT)orange.fr), May 01 2002

Expanding x/cosh(x) gives alt. signed values at odd positions.

a(n)=(2n+1)*A000364(n)=sum(i=0, 2n, B(i)*C(2n+1, i)*4^i)=(2n+1)*E(2n) where B(i) are the Bernoulli numbers, C(2n, i) the binomial numbers and E(2n) the Euler numbers. - Benoit Cloitre (benoit7848c(AT)orange.fr), May 01 2002

Recurrence: a(n) = -(-1)^n*Sum[i=0..n-1, (-1)^i*a(i)*C(2n+1, 2i+1) ]. - Ralf Stephan, Feb 24 2005

seq((2*i+1)!*coeff(convert(series(x/cos(x), x, 32), polynom), x, 2*i+1), i=0..13); [From Peter Luschny (peter(AT)luschny.de), May 03 2009]

x/Cos[ x ] (* Odd Part *)

(PARI) for(n=0, 25, print1(sum(i=0, 2*n+1, binomial(2*n+1, i)*bernfrac(i)*4^i), ", "))

Bisection of A009391, A009392, A065619, A083008.

Cf. A099028.

Sequence in context: A143925 A074708 A160143 this_sequence A136173 A003024 A131310

Adjacent sequences: A009840 A009841 A009842 this_sequence A009844 A009845 A009846

nonn

R. H. Hardin (rhhardin(AT)att.net)

Extended and signs tested Mar 15 1997 by Olivier Gerard.