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%I #17 Aug 18 2018 16:26:07
%S 1,1,2,6,22,98,514,3110,21334,163650,1388162,12902086,130391830,
%T 1423632546,16699055490,209432697830,2796597560150,39613075175554,
%U 593253347702530,9366042608039814,155466234198142998
%N Expansion of o.g.f.: Sum_{n>=0} Product_{k=1..n} tanh(k*arctanh(x)).
%H Vaclav Kotesovec, <a href="/A177389/b177389.txt">Table of n, a(n) for n = 0..280</a>
%F O.g.f.: A(x) = Sum_{n>=0} A002105(n+1)*arctanh(x)^n/n!, where A002105 is the reduced tangent numbers.
%F G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - (1-x)^k)/((1+x)^k + (1-x)^k). - _Paul D. Hanna_, May 22 2010
%F a(n) ~ 2^(3*n+9/2) * n^(n+1) / (exp(n) * Pi^(2*n+2)). - _Vaclav Kotesovec_, Nov 06 2014
%e O.g.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 98*x^5 + 514*x^6 + ...
%e Let G(x) = Sum_{n>=0} A002105(n+1)*x^n/n!, so that
%e G(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 496*x^4/4! + 11056*x^5/5! + ...
%e then A(x) = G(arctanh(x)).
%e G.f.: 1 + x + x*(2x/(1+x^2)) + x*(2x/(1+x^2))*((3x+x^3)/(1+3x^2)) + x*(2x/(1+x^2))*((3x+x^3)/(1+3x^2))*((4x+4x^3)/(1+6x^2+x^4)) + ... - _Paul D. Hanna_, May 22 2010
%o (PARI) {a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,tanh(k*atanh(X))));polcoeff(Egf,n)}
%o (PARI) {a(n)=polcoeff(sum(m=0,n,prod(k=1,m,((1+x)^k-(1-x)^k)/((1+x)^k+(1-x)^k+x*O(x^n)))),n)} \\ _Paul D. Hanna_, May 22 2010
%Y Cf. A002105.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 15 2010
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