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A009710 E.g.f. tan(tan(x)^2) (even powers only). 2

%I #18 Jan 24 2015 04:35:16

%S 0,2,16,512,34816,3821312,618121216,138682959872,41171702972416,

%T 15610723195092992,7357121913006063616,4217775794187229724672,

%U 2889975739296119171055616,2332177121915783600628826112

%N E.g.f. tan(tan(x)^2) (even powers only).

%F a(n)=4*sum(m=0..n-1/2, ((sum(j=1..2*m+1, j!*2^(2*m-j)*(-1)^(m+1+j)*stirling2(2*m+1,j)))*sum(j=4*m+2..2*n, binomial(j-1,4*m+1)*j!*2^(2*n-j-1)*(-1)^(n+1+j)*stirling2(2*n,j)))/(2*m+1)!). [Vladimir Kruchinin, Jun 21 2011]

%F a(n) ~ (2*n)! * 2 * sqrt(2) / ((2+Pi) * sqrt(Pi) * arctan(sqrt(Pi/2))^(2*n+1)). - _Vaclav Kotesovec_, Jan 24 2015

%e tan(tan(x)*tan(x))=2/2!*x^2+16/4!*x^4+512/6!*x^6+34816/8!*x^8...

%t nn = 20; Table[(CoefficientList[Series[Tan[Tan[x]^2], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* _Vaclav Kotesovec_, Jan 24 2015 *)

%o (Maxima)

%o a(n):=4*sum(((sum(j!*2^(2*m-j)*(-1)^(m+1+j)*stirling2(2*m+1,j),j,1,2*m+1))*sum(binomial(j-1,4*m+1)*j!*2^(2*n-j-1)*(-1)^(n+1+j)*stirling2(2*n,j),j,4*m+2,2*n))/(2*m+1)!,m,0,n-1/2); [Vladimir Kruchinin, Jun 21 2011]

%K nonn

%O 0,2

%A _R. H. Hardin_

%E Extended and signs tested Mar 15 1997 by _Olivier GĂ©rard_.

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Last modified October 4 16:17 EDT 2023. Contains 365887 sequences. (Running on oeis4.)