%I #18 Oct 13 2017 05:12:39

%S 1,1,2,4,4,8,202,-1024,-17718,590262,-4319954,-250042344,11097553206,

%T -110198613306,-12018093009424,791558086255136,-14143565415027956,

%U -1546249362315735600,162730885462262914406,-5686938689391780668440,-410162666115593769228098,77767801528714637765717294

%N E.g.f. satisfies: A(exp(x) - 1) = x*exp(A(x)).

%F a(n) = n!*T(n,1), T(n,m) = Sum_{k=1..n-m} (T(n-m, k)*m^k/k! - Stirling2(n, k+m-1)*((k+m-1)!/n!)*T(k+m-1, m)), n > m, with T(n,n) = 1.

%F E.g.f. satisfies A(x) = log(1+x)*exp( A(log(1+x)) ).

%e E.g.f.: A(x) = x + x^2/2! + 2*x^3/3! + 4*x^4/4! + 4*x^5/5! + 8*x^6/6! + 202*x^7/7! + ...

%e A(exp(x)-1) = x + x^2 + x^3 + x^4 + 11/12*x^5 + 3/4*x^6 + 107/180*x^7 + 59/120*x^8+ ...

%e x*exp(A(x)) = x + x^2 + x^3 + x^4 + 11/12*x^5 + 3/4*x^6 + 107/180*x^7 + 59/120*x^8+ ...

%t t[n_, m_] := t[n, m] = If[ n == m , 1 , Sum[t[n-m, k]*m^k/k! - StirlingS2[n, k+m-1]*(k+m-1)!/n!*t[k+m-1, m], {k, 1, n-m}]]; a[n_] := n!*t[n, 1]; Table[a[n], {n, 1, 22}] (* _Jean-François Alcover_, Jun 24 2013, after _Vladimir Kruchinin_ *)

%o (Maxima) T(n,m):=if n=m then 1 else sum(T(n-m,k)*m^k/k!-stirling2(n,k+m-1)*(k+m-1)!/n!*T(k+m-1,m),k,1,n-m);

%o makelist(n!*T(n,1),n,1,15);

%o (PARI) /* Using A(exp(x)-1) = x*exp(A(x)) */

%o {a(n)=local(A=x);for(i=1,n,A=log(1+x+x*O(x^n))*exp(subst(A,x,log(1+x+x*O(x^n)))));n!*polcoeff(A,n)} \\ _Paul D. Hanna_

%o (PARI) /* Using _Vladimir Kruchinin_'s formula: */

%o {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}

%o {T(n,k)=if(n<k,0,if(n==k,1,sum(j=1,n-k,T(n-k,j)*k^j/j!-Stirling2(n,k+j-1)*(k+j-1)!/n!*T(k+j-1,k)) ))}

%o {a(n)=n!*T(n,1)} \\ _Paul D. Hanna_

%K sign

%O 1,3

%A _Vladimir Kruchinin_, Dec 04 2011