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%I M3144 #36 Oct 02 2021 07:56:58
%S 1,3,37,1015,47881,3459819,354711853,48961863007,8754050024209,
%T 1967989239505875,543326939019354421,180718022989699819207,
%U 71275877445849484090393,32890432371345908634652347,17555593768891213894861569085
%N Expansion of e.g.f. tan(sinh(x)) (odd powers only).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A003716/b003716.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = Sum_{k=1..n} ((-1)^(k-1)+1)/(2^k*k!) * ( Sum_{i=0..k} (-1)^i*(k-2*i)^n *binomial(k,i) ) * ( Sum_{j=1..k} j! * 2^(k-j-1) * (-1)^((k+1)/2+j) * stirling2(k,j) ). - _Vladimir Kruchinin_, Apr 20 2011
%F a(n) ~ 4 * (2*n+1)! / (sqrt(4+Pi^2) * (log((Pi + sqrt(4+Pi^2))/2))^(2*n+2)). - _Vaclav Kotesovec_, Feb 16 2015
%t Tan[ Sinh[ x ] ] (* Odd Part *)
%t nn = 20; Table[(CoefficientList[Series[Tan[Sinh[x]], {x, 0, 2*nn+1}], x] * Range[0, 2*nn+1]!)[[n]], {n, 2, 2*nn, 2}] (* _Vaclav Kotesovec_, Feb 16 2015 *)
%o (Maxima)
%o a(n):=sum(((-1)^(k-1)+1)/(2^k*k!)*sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k)*(sum(j!*2^(k-j-1)*(-1)^((k+1)/2+j)*stirling2(k,j),j,1,k)),k,1,n); [_Vladimir Kruchinin_, Apr 20 2011]
%Y Cf. A003718, A003721.
%K nonn
%O 0,2
%A _R. H. Hardin_
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