OFFSET
1,1
COMMENTS
Related to those formulas derived from Bernoulli polynomials: Sum_{k>0} sin(k*x)/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).
LINKS
FORMULA
From Peter Bala, Mar 02 2015: (Start)
a(n) = 6^(2*n - 1)*B(2*n - 1,1/6), where B(n,x) denotes the n-th Bernoulli polynomial. Cf. A002111, A009843, A069852.
Conjecturally, a(n) = 2 * the unsigned numerator of B(2*n - 1,1/6). If true then this sequence is a bisection of 2*A158073.
G.f.: -3*t*sinh(2*t)/sinh(3*t) = -2*t + 10*t^3/3! - 170*t^5/5! + ....
G.f.: Sum_{n >= 0} { 2/(n+1) * Sum_{k = 0..n} (-1)^(k+1)*binomial(n,k)/( (1 - (6*k + 1)*x)*(1 - (6*k + 5)*x) ) } = -2 + 10*x^2 - 170*x^4 + 6370*x^6 - ....
(End)
MAPLE
seq(6^(2*n-1)*bernoulli(2*n-1, 1/6), n=1..14); # (after Peter Bala) Peter Luschny, Mar 08 2015
PROG
(PARI) for(n=1, 25, print1(sum(i=0, 2*n, binomial(2*n+1, i)*bernfrac(i)*6^i), ", "))
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Benoit Cloitre, May 01 2002
STATUS
approved