%I #18 Nov 28 2023 08:50:05
%S 1,1,1,7,25,121,841,9871,80977,869905,10776241,131366071,1821918121,
%T 27671299657,460068491065,8716820294911,162728020119841,
%U 3217989767498401,69343322972016097,1533322325194196455
%N E.g.f. satisfies A(x) = exp(x*A(x^2)).
%H Vaclav Kotesovec, <a href="/A138292/b138292.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = T(2*n+1), where T(n,m) = (1+(-1)^(n-m))/2*((n-m)/2)!*sum(k=1..(n-m)/2, m^k*T((n-m)/2,k)/(k!*((n-m-2*k)/4)!)), n>m, T(n,n)=1. - _Vladimir Kruchinin_, Mar 18 2015
%F a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/2)} (2*k+1) * a(k) * a(n-1-2*k) / (k! * (n-1-2*k)!). - _Seiichi Manyama_, Nov 28 2023
%e E.g.f: A(x) = 1 + x + 1/2*x^2 + 7/6*x^3 + 25/24*x^4 + 121/120*x^5 +...
%e Log(A(x)) = x + x^3 + 1/2*x^5 + 7/6*x^7 + 25/24*x^9 + 121/120*x^11 +...
%o (PARI) {a(n)=local(A=1); for(i=0,n-1,A=exp(x*subst(A,x,x^2+x*O(x^n))));n!*polcoeff(A,n)}
%o (Maxima)
%o T(n,m):=if n=m then 1 else (1+(-1)^(n-m))/2*((n-m)/2)!*sum(m^k*T((n-m)/2,k)/(k!*((n-m-2*k)/4)!),k,1,(n-m)/2);
%o makelist(T(2*n-1,1),n,1,20); /* _Vladimir Kruchinin_, Mar 18 2015 */
%K nonn
%O 0,4
%A _Paul D. Hanna_, Mar 13 2008