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%I #15 Oct 10 2017 02:03:53
%S 1,3,26,376,7614,198248,6309092,237291388,10297903920,506495785632,
%T 27842563031304,1691646018671376,112569103111005072,
%U 8142200129607522288,636046143210331062048,53366672768969064921024
%N E.g.f.: A(x) = -log(1 - x - A(x)^2).
%F a(n) = Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} ((-1)^(n+j-1)/(k-j)!)*Sum_{L=0..min(j, (n+j-1)/2)} Stirling2(n-2*L+j-1, j-l)/(L!*(n-2*L+j-1)!), n > 0. - _Vladimir Kruchinin_, Feb 03 2012
%F a(n) ~ n^(n-1) / (sqrt(2*(1+c)) * exp(n) * (1-2*c-c^2)^(n-1/2)), where c = LambertW(1/2). - _Vaclav Kotesovec_, Dec 28 2013
%e A(x) = x + 3*x^2/2! + 26*x^3/3! + 376*x^4/4! + 7614*x^5/5! +...
%e x + A(x)^2 = 1 - exp(-A(x)) = G(x) = g.f. of A143154:
%e G(x) = x + 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! +...
%e A(x)^2 = 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! +...
%t Table[Sum[(n+k-1)!*Sum[(-1)^(n+j-1)/(k-j)!*Sum[(StirlingS2[n-2*l+j-1,j-l])/(l!*(n-2*l+j-1)!),{l,0,Min[j,(n+j-1)/2]}],{j,0,k}],{k,0,n-1}],{n,1,20}] (* _Vaclav Kotesovec_ after _Vladimir Kruchinin_, Dec 28 2013 *)
%o (PARI) {a(n)=local(A=x+O(x^n));for(i=0,n,A=-log(1-x-A^2));n!*polcoeff(A,n)}
%o (PARI) {a(n)=n!*polcoeff(-log(1-serreverse(x-log(1-x+x*O(x^n))^2)),n)}
%o (Maxima) a(n):=(sum((n+k-1)!*sum((-1)^(n+j-1)/(k-j)!*sum((stirling2(n-2*l+j-1,j-l))/(l!*(n-2*l+j-1)!),l,0,min(j,(n+j-1)/2)),j,0,k),k,0,n-1)); /* _Vladimir Kruchinin_, Feb 03 2012 */
%Y Cf. A143139, A143154, A202356.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 27 2008
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