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%I #19 Oct 13 2017 05:06:11
%S 1,1,2,12,108,1360,21780,424998,9774912,259012080,7769656800,
%T 260283596760,9631680917760,390185658289128,17175153440774784,
%U 816267894739416000,41658264473400852480,2272233977181361580160,131913883517800157429760,8121310193676734923381056
%N E.g.f. satisfies: A(x) = 1 + x*A(x)^A(x).
%F E.g.f.: 1 + Series_Reversion( x * Sum_{n>=0} (-x)^n/n! * Product_{k=1..n} (k+x) ). - _Paul D. Hanna_, Sep 27 2014
%F a(n) = n!*Sum_{k=0..n-1} ((Sum_{i=k..n-1} (Stirling1(i,k)*binomial(k, n-i-1)/i!))*n^(k-1)), n > 0, a(0)=1. - _Vladimir Kruchinin_, Jan 24 2012
%F a(n) ~ n^(n-1) * (s-1)*sqrt(s/(1+(s-1)*s)) / (exp(n)*r^n), where s = 1.662886128060660201... is the root of the equation (s-1)*(1+log(s)) = 1, and r = (s-1)/s^s = 0.2845572964785024040... . - _Vaclav Kotesovec_, Jul 15 2014
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1360*x^5/5! + ...
%e where A(x)^A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 272*x^4/4! + 3630*x^5/5! + ...
%e The e.g.f. also satisfies:
%e (1/x)*Series_Reversion(A(x) - 1) = 1 - x*(1+x) + x^2*(1+x)*(2+x)/2! - x^3*(1+x)*(2+x)*(3+x)/3! + x^4*(1+x)*(2+x)*(3+x)*(4+x)/4! - x^5*(1+x)*(2+x)*(3+x)*(4+x)*(5+x)/5! +- ...
%t Flatten[{1,Table[n!*Sum[Sum[StirlingS1[i,k]*Binomial[k,n-i-1]/i!*n^(k-1),{i,k,n-1}],{k,0,n-1}],{n,1,20}]}] (* _Vaclav Kotesovec_, Jul 15 2014 after _Vladimir Kruchinin_ *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*(A+x*O(x^n))^A);n!*polcoeff(A,n)}
%o (Maxima) a(n):=if n=0 then 1 else (n!*sum((sum((stirling1(i,k)*binomial(k,n-i-1))/i!,i,k,n-1))*n^(k-1),k,0,n-1)); /* _Vladimir Kruchinin_, Jan 24 2012 */
%K nonn
%O 0,3
%A _Paul D. Hanna_, Sep 02 2011
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