%I #3 Mar 30 2012 18:36:43
%S 1,1,3,-3,-57,369,3861,-76617,-413775,26879103,-52349229,-13272657111,
%T 163271276343,8454393401121,-248550519324411,-5938622542044441,
%U 389600359347114081,2878180608333977343,-677734293694386004317,5701654525891937630409
%N E.g.f. A(x) satisfies: Sum_{k=0..n} (A(x)^n)_k/k! = [exp(n)] (integer floor of e^n) for all n>=0, where (A(x)^n)_k/k! is the coefficient of x^k in A(x)^n.
%C See triangle A100064 of initial coefficients of successive powers of the e.g.f. for this sequence.
%e List the coefficients of powers of e.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!:
%e A(x)^0: [1,__0,0,0,0,0,0,0,0,...],
%e A(x)^1: [1,1,__3,-3,-57,369,3861,-76617,-413775,...],
%e A(x)^2: [1,2,8,__12,-84,-12,7200,-40716,-1301328,...],
%e A(x)^3: [1,3,15,51,__27,-513,4077,33237,-1211895,...],
%e A(x)^4: [1,4,24,120,408,__216,-3168,45576,-202176,...],
%e A(x)^5: [1,5,35,225,1215,4365,__1485,-27765,440865,...],...
%e then for each row n, Sum_{k=0..n} (A(x)^n)_k/k! = [exp(n)]:
%e [exp(0)] = 1 = 1
%e [exp(1)] = 1+1 = 2
%e [exp(2)] = 1+2+8/2! = 7
%e [exp(3)] = 1+3+15/2!+51/3! = 20
%e [exp(4)] = 1+4+24/2!+120/3!+408/4! = 54
%e [exp(5)] = 1+5+35/2!+225/3!+1215/4!+4365/5! = 148
%o (PARI) {a(n)=local(A,C,F,G);if(n==0,A=1,F=sum(k=0,n-1,a(k)*x^k/k!); C=floor(exp(n))-sum(k=0,n-1,polcoeff(F^n+x*O(x^k),k,x)); G=sum(k=0,n-1,polcoeff(F^n+x*O(x^k),k,x)*x^k); A=n!*polcoeff((G+C*x^n)^(1/n)+x*O(x^n),n,x));A}
%Y Cf. A100064.
%K sign
%O 0,3
%A _Paul D. Hanna_, Nov 02 2004