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A213357 E.g.f. satisfies A(x) = 1 + (exp(x) - 1) * A(exp(x) - 1).

%I A213357 #22 Jun 04 2022 09:42:15
%S A213357 1,1,3,16,133,1561,24374,485640,11969843,356348290,12572687675,
%T A213357 517644938724,24553141710156,1327223189312606,81005220402829714,
%U A213357 5537660009982114858,421050946315817655785,35387457515051683169307,3269500807582223015227780
%N A213357 E.g.f. satisfies A(x) = 1 + (exp(x) - 1) * A(exp(x) - 1).
%H A213357 Vincenzo Librandi, <a href="/A213357/b213357.txt">Table of n, a(n) for n = 0..200</a>
%F A213357 a(n) = Sum_{k=1..n} k * a(k-1) * Stirling2(n, k) if n>0.
%F A213357 A048801(n) = n * a(n-1) = Sum_{k=1..n} a(k) * Stirling1(n, k) if n>0.
%e A213357 1 + x + 3*x^2 + 16*x^3 + 133*x^4 + 1561*x^5 + 24374*x^6 + 485640*x^7 + ...
%t A213357 nmax=20; b = ConstantArray[0,nmax+1]; b[[1]]=1; Do[b[[n+1]] = Sum[k*b[[k]]*StirlingS2[n, k],{k,1,n}],{n,1,nmax-1}]; b (* _Vaclav Kotesovec_, Mar 12 2014 *)
%o A213357 (PARI) {a(n) = local(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = subst( 1 + x * A, x, exp( x + x * (A - A)) - 1)); n! * polcoeff( A, n))}
%o A213357 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*stirling(i, j, 2)*v[j])); v; \\ _Seiichi Manyama_, Jun 04 2022
%Y A213357 Cf. A048801, A135750.
%K A213357 nonn
%O A213357 0,3
%A A213357 _Michael Somos_, Jun 09 2012