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%I #17 Apr 17 2017 12:49:45
%S 1,4,76,3424,277456,35345344,6504742336,1632531979264,535821754153216,
%T 222769351470429184,114411762387714436096,71132353206363509039104,
%U 52648938670226334981246976
%N Expansion of tan(arcsin(sinh(x))) (odd powers only).
%H G. C. Greubel, <a href="/A012101/b012101.txt">Table of n, a(n) for n = 0..219</a>
%F a(n) = Sum_{m=0..n} ( binomial(2*m,m)*2^(-4*m)*( Sum_{i=0,..,(2*m+1)/2} (2*i-2*m-1)^(2*n+1)*binomial(2*m+1,i)*(-1)^(i+1) ) ). - _Vladimir Kruchinin_, Jun 15 2011
%F E.g.f.: sinh(x) / sqrt(1 - sinh(x)^2). - _Vaclav Kotesovec_, Feb 06 2015
%F a(n) ~ (2*n+1)! / (sqrt(Pi*n) * 2^(1/4) * (log(1+sqrt(2)))^(2*n+3/2)). - _Vaclav Kotesovec_, Feb 06 2015
%e tan(arcsin(sinh(x))) = x+4/3!*x^3+76/5!*x^5+3424/7!*x^7+277456/9!*x^9...
%t nn = 20; Table[(CoefficientList[Series[Sinh[x]/Sqrt[1 - Sinh[x]^2], {x, 0, 2*nn+1}], x] * Range[0, 2*nn+1]!)[[n]], {n, 2, 2*nn, 2}] (* _Vaclav Kotesovec_, Feb 06 2015 *)
%t Table[Sum[Binomial[2*m, m]*2^(-4*m)*Sum[(2*i - 2*m - 1)^(2*n + 1)*
%t Binomial[2*m + 1, i]*(-1)^(i + 1), {i, 0, (2*m + 1)/2}], {m, 0, n}], {n, 0, 50}] (* _G. C. Greubel_, Feb 15 2017 *)
%o (Maxima)
%o a(n):=sum(binomial(2*m,m)*2^(-4*m)*sum((2*i-2*m-1)^(2*n+1)*binomial(2*m+1,i)*(-1)^(i+1),i,0,(2*m+1)/2),m,0,n); /* _Vladimir Kruchinin_, Jun 15 2011 */
%Y Cf. A012571.
%K nonn
%O 0,2
%A Patrick Demichel (patrick.demichel(AT)hp.com)
%E Typo in second formula corrected (following a suggestion of _Sergei N. Gladkovskii_) by _Vaclav Kotesovec_, Apr 17 2017
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