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%I #11 Jan 22 2014 15:48:48
%S 1,-2,9,-68,725,-9966,167629,-3334120,76543785,-1992009850,
%T 57948521521,-1863394764780,65631109286717,-2512768138160294,
%U 103905545328667125,-4615035074291158352,219122841820491458897,-11075488610594107402098,593746153204862664363481
%N E.g.f. is series reversion of x*(sec(x)+tan(x)).
%F a(n) = sum(k=1..n-1, (-1)^k*binomial(n+k-1,n-1)*sum(k=1..n-1, binomial(n-2*i-2,k-1)*sum(i=0..(n-k-1)/2, (-1)^(j+i)*2^(-n+k-j+2*i+1)*stirling2(n-1,n+j+(-2)*i-1)*binomial(n+j+(-2)*i-2,n-2*i-2)*(n+j+(-2)*i-1)!,j,0,2*i))), n>1, a(1)=1.
%F a(n) ~ (-1)^(n+1) * n^(n-1) * s / (sqrt(1+sin(s)) * exp(n) * (1-sin(s))^n), where s = 0.73908513321516... (see A003957) is the root of the equation s = cos(s). - _Vaclav Kotesovec_, Jan 22 2014
%t a[n_] := Sum[ (-1)^k*Binomial[n+k-1, n-1] * Sum[ Binomial[n-2*i-2, k-1] * Sum[ (-1)^(j+i)*2^(-n+k-j+2*i+1)*StirlingS2[n-1, n+j-2*i-1] * Binomial[n+j-2*i-2, n-2*i-2]*(n+j-2*i-1)!, {j, 0, 2*i}], {i, 0, (n-k-1)/2}], {k, 1, n-1}]; a[1] = 1; Table[a[n], {n, 1, 19}] (* _Jean-François Alcover_, Feb 22 2013 *)
%t Rest[CoefficientList[InverseSeries[Series[x*(Sec[x]+Tan[x]), {x, 0, 20}], x],x]*Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 22 2014 *)
%o (Maxima) a(n):=if n=1 then 1 else
%o sum((-1)^k*binomial(n+k-1,n-1)*sum(binomial(n-2*i-2,k-1)*sum((-1)^(j+i)*2^(-n+k-j+2*i+1)*stirling2(n-1,n+j+(-2)*i-1)*binomial(n+j+(-2)*i-2,n-2*i-2)*(n+j+(-2)*i-1)!,j,0,2*i),i,0,(n-k-1)/2),k,1,n-1)
%Y Cf. A003957.
%K sign
%O 1,2
%A _Vladimir Kruchinin_, Sep 26 2012
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