%I #10 Mar 31 2012 10:23:14
%S 1,0,2,0,24,0,552,0,15936,0,427008,0,35567232,0,7352654592,0,
%T -1064016089088,0,-285531029213184,0,515396596477280256,0,
%U -177833505179096580096,0,-286336846462160593059840,0,573948928392687509387673600,0,-211350462543692217044652785664
%N E.g.f. satisfies: A(sinh(x)) = x*cosh(A(x)).
%F a(n)=T(n,1), T(n,m)=sum(k=1..n-m, T(n-m,k)*sum(i=0..m, (m-2*i)^k*binomial(m,i))/(k!*2^m)-(2^(-m-k+1)*sum(i=0..m+k-1, (-1)^i*(m+k-2*i-1)^n*binomial(m+k-1,i)))/n!*T(k+m-1,m))), n>m, with T(n,n)=1.
%F E.g.f. satisfies: A(x) = log(sqrt(1+x^2) + x) * cosh( A( log(sqrt(1+x^2) + x) ) ). - Paul D. Hanna, Dec 06 2011
%e A(x) = x +x^3/3 +x^5/5 +23*x^7/210 + 83*x^9/1890 +...
%o (Maxima)
%o array(B,100,100);
%o fillarray (B, makelist (-1, i, 1, 10000));
%o T(n,m):=if B[n,m]=-1 then B[n,m]:(if n=m then 1 else sum(T(n-m,k)*sum((m-2*i)^k*binomial(m,i),i,0,m)/(k!*2^m)-(2^(-m-k+1)*sum((-1)^i*(m+k-2*i-1)^n*binomial(m+k-1,i),i,0,m+k-1))/n!*T(k+m-1,m),k,1,n-m)) else B[n,m];
%o makelist(n!*T(n,1),n,1,30);
%o (PARI) /* Using A(x) = arcsinh(x)*cosh(A(arcsinh(x))), Paul D. Hanna, Dec 06 2011 */
%o {a(n)=local(A=1+x,X=x+x*O(x^n));for(i=1,n+1,A=log(sqrt(1+X^2)+x)*cosh(subst(A,x,log(sqrt(1+X^2)+x))));n!*polcoeff(A,n)}
%K sign
%O 1,3
%A _Vladimir Kruchinin_, Dec 06 2011