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%I #15 Apr 14 2024 11:38:49
%S 0,1,2,13,164,3355,100886,4185097,228970568,15972720439,1383706615610,
%T 145736540156581,18339615566386412,2717605030233712723,
%U 468371974894477377374,92895125380418204480065,21008128723110866359626896,5373571097376355083238621807
%N a(n)=(-1)^(n+1)*(3/4)*(9^n-1)*B(2n) where B(k) denotes the k-th Bernoulli number.
%F a(n) = (2n)! [x^(2n)] (3/2) x sin(x)/(2 cos(x)+1). - Ira M. Gessel Feb 23 2012
%F a(n) = (-1)^n*(Sum_{i, 0, 2*n - 1} (Bernoulli(i)*binomial(2*n, i)*3^i))/2. - _Detlef Meya_, Apr 14 2024
%F a(n) ~ sqrt(Pi) * 3^(2*n+1) * n^(2*n + 1/2) / (Pi^(2*n) * exp(2*n)). - _Vaclav Kotesovec_, Apr 14 2024
%t a[n_]:=(-1)^n*Sum[BernoulliB[i]*Binomial[2*n,i]*3^i,{i,0,2*n-1}]/2; Flatten[Table[a[n],{n,0,17}]] (* _Detlef Meya_, Apr 14 2024 *)
%o (PARI) a(n)=(-1)^(n+1)*(3/4)*(9^n-1)*bernfrac(2*n)
%K nonn,changed
%O 0,3
%A _Benoit Cloitre_, Dec 13 2003
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