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A209519 Expansion A(x) = Sum_{n>0} a(n)*x^n/(3^(n-1)*n!), A(x) satisfies A(A(A(x)))=e^x-1.

%I A209519 #12 May 12 2024 10:04:08
%S A209519 1,1,0,0,2,-21,138,150,-22833,303975,3451320,-214016553,666006714,
%T A209519 228865308144,-4943013567642,-396567325158381,21423378444873687,
%U A209519 1022158819761317838,-121532275123709160942
%N A209519 Expansion A(x) = Sum_{n>0} a(n)*x^n/(3^(n-1)*n!), A(x) satisfies A(A(A(x)))=e^x-1.
%H A209519 Seiichi Manyama, <a href="/A209519/b209519.txt">Table of n, a(n) for n = 1..200</a>
%F A209519 a(n)=3^(n-1)*n!*T(n,1), T(n,m)=1/3*(stirling2(n,m)*m!/n!-sum(k=m+1..n-1, T(k,m)*sum(i=k..n, T(n,i)*T(i,k)))-T(m,m)*sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1.
%o A209519 (Maxima)
%o A209519 T(n,m):=if n=m then 1 else 1/3*(stirling2(n,m)*m!/n!-sum(T(k,m)*sum(T(n,i)*T(i,k),i,k,n),k,m+1,n-1)-T(m,m)*sum(T(n,i)*T(i,m),i,m+1,n-1));
%o A209519 makelist(n!*3^(n-1)*(T(n,1)),n,1,7);
%Y A209519 Cf. A184011.
%K A209519 sign
%O A209519 1,5
%A A209519 _Vladimir Kruchinin_, Mar 10 2012