OFFSET
1,1
FORMULA
a(n)=2*(sum(m=2..2*n, ((sum(k=1..m-1, binomial(k,m-k-1)*(-1)^(k+1)*3^(2*n-2*m+k+1)*binomial(m+k-1,m-1)))*sum(j=1..m, ((sum(i=0..((j-1)/2), (j-2*i)^(2*n)*binomial(j,i)))*binomial(m,j)*(-1)^(n+m-j))/2^j))/(m)))-
((-1)^n*3^(2*n-1)), n>0.
a(n) ~ Gamma(1/3) * 2^(4*n - 2/3) * 3^(2*n - 1/2) * n^(2*n - 5/6) / (Pi^(2*n + 1/6) * exp(2*n)). - Vaclav Kotesovec, Jun 05 2019
MATHEMATICA
nmax = 40; Table[(CoefficientList[Series[1 - Cos[3*x]^(1/3), {x, 0, nmax}], x] * Range[0, nmax]!)[[n]], {n, 3, nmax, 2}] (* Vaclav Kotesovec, Jun 05 2019 *)
PROG
(Maxima)
a(n):=2*(sum(((sum(binomial(k, m-k-1)*(-1)^(k+1)*3^(2*n-2*m+k+1)*binomial(m+k-1, m-1), k, 1, m-1))*sum(((sum((j-2*i)^(2*n)*binomial(j, i), i, 0, ((j-1)/2)))*binomial(m, j)*(-1)^(n+m-j))/2^j, j, 1, m))/(m), m, 2, 2*n))-
((-1)^n*3^(2*n-1));
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Jun 05 2011
STATUS
approved