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%I #11 Jun 05 2019 09:41:06
%S 3,27,1863,259767,63267723,23850461907,12872337567183,
%T 9418588525038447,8974900856105748243,10799459611549296021387,
%U 16014456358054037241378903,28692834058049011948073522727,61105982516981628849258186347163,152570799245287136693700721604134467,441413217492406160002632205611608461023
%N E.g.f. 1 - cos(3*x)^(1/3). (even powers only)
%F 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)))-
%F ((-1)^n*3^(2*n-1)), n>0.
%F 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
%t 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 *)
%o (Maxima)
%o 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))-
%o ((-1)^n*3^(2*n-1));
%K nonn
%O 1,1
%A _Vladimir Kruchinin_, Jun 05 2011
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