%I #16 Nov 29 2017 04:30:08
%S 1,3,9,28,90,300,1051,3975,16971,86584,550560,4354308,41245021,
%T 448722207,5443128597,72294557416,1039558994214,16059538853232,
%U 264996063891607,4648786414414347,86363450200625247,1693336666564186012
%N G.f. A(x) satisfies: A(x) = (1-x)^3*A(x)^2 - x^2*A'(x).
%H Vincenzo Librandi, <a href="/A143739/b143739.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) ~ (n-1)! / 2. - _Vaclav Kotesovec_, Feb 22 2014
%F for n > 1, a(n) = (n+1)*(n+2)/2 + (n-1)*a(n-1) + Sum_{k=1..n-1}(-a(k) + (k+1)*(k+2)/2)*a(n-k) + (k+1)*(k+2)*(n-1-k)*a(n-1 - k)/2. - _Tani Akinari_, Nov 29 2017
%e G.f.: A(x) = 1 + 3*x + 9*x^2 + 28*x^3 + 90*x^4 + 300*x^5 + 1051*x^6 + ...
%e A(x)^2 = 1 + 6*x + 27*x^2 + 110*x^3 + 429*x^4 + 1644*x^5 + 6306*x^6 + ...
%e (1-x)^3*A(x)^2 = 1 + 3*x + 12*x^2 + 46*x^3 + 174*x^4 + 660*x^5 + ...
%e x^2*A'(x) = 3*x^2 + 18*x^3 + 84*x^4 + 360*x^5 + 1500*x^6 + 6306*x^7 + ...
%t Table[Coefficient[Series[2*E^x/(E^x*(2 + (-5 + 1/x)*(1/x)) - ExpIntegralEi[x]), {x, Infinity, 20}], x, -n], {n, 0, 20}] (* _Vaclav Kotesovec_, Feb 22 2014 *)
%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x^2*deriv(A)/A)/(1-x)^3); polcoeff(A, n)}
%o (Maxima) a[n]:=if n<2 then 2*n+1 else (n+1)*(n+2)/2+(n-1)*a[n-1]+sum((-a[k]+(k+1)*(k+2)/2)*a[n-k]+(k+1)*(k+2)*(n-1-k)*a[n-1-k]/2,k,1,n-1);
%o makelist(a[n],n,0,100); /* _Tani Akinari_, Nov 29 2017 */
%Y Cf. A137553 (variant).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 05 2008