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%I #10 Dec 21 2017 17:45:40
%S 0,1,-2,18,-312,9470,-436860,28616322,-2522596496,288046961190,
%T -41355026494020,7291524732108650,-1548849359704927896,
%U 390122366308850972238,-114968364853645904762252,39189956630839558368115410,-15300235972710835734174638880
%N Expansion of e.g.f. log(1 + x*tan(x/2)) (even powers only).
%H Vaclav Kotesovec, <a href="/A296837/b296837.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = (2*n)! * [x^(2*n)] log(1 + x*tan(x/2)).
%F a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.54340463841820844795870974005331555369788376471926269... is the root of the equation r*tanh(r/2) = 1. - _Vaclav Kotesovec_, Dec 21 2017
%e log(1 + x*tan(x/2)) = x^2/2! - 2*x^4/4! + 18*x^6/6! - 312*x^8/8! + ...
%t nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
%Y Cf. A001469, A003707, A009379, A009399, A110501, A296838.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Dec 21 2017
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