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%I #13 Mar 16 2024 17:16:07
%S 0,1,1,33,437,22205,978873,81005113,7356832669,949918117653,
%T 142805534055905,27120922891214801,6016195462632487941,
%U 1592800634594574194413,486576430503128985793417,171866951067212728072402665,69025662074064538734826793453
%N Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).
%H Vaclav Kotesovec, <a href="/A296839/b296839.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = (2*n)! * [x^(2*n)] tan(x*tan(x/2)).
%F a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 16/Pi^2 = 1.621138938277404343102071411355642222469740394755... is the root of the equation tan(1/sqrt(d)) = Pi*sqrt(d)/4 and c = 1.75568815831... - _Vaclav Kotesovec_, Dec 21 2017, updated Mar 16 2024
%e tan(x*tan(x/2)) = x^2/2! + x^4/4! + 33*x^6/6! + 437*x^8/8! + ...
%t nmax = 16; Table[(CoefficientList[Series[Tan[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
%Y Cf. A000182, A001469, A003718, A009707, A024265, A110501, A296835, A296841, A296842.
%K nonn
%O 0,4
%A _Ilya Gutkovskiy_, Dec 21 2017
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