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%I #16 Feb 24 2014 02:02:08
%S 1,1,6,91,2910,187178,24019884,6154080275,3151538898870,
%T 3227331249742334,6609648919647088788,27073195436180090799006,
%U 221783764770326660974008300,3633705802215756626623500731892,119069276624759801067298501607804376
%N G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n/n! * d^n/dx^n x*A(x)^2.
%H Vincenzo Librandi, <a href="/A182263/b182263.txt">Table of n, a(n) for n = 0..70</a>
%F a(n) = (2^n-1) * { [x^(n-1)] A(x)^2 } for n>0 with a(0)=1.
%F a(n) = (2^n-1) * Sum_{k=0..n-1} a(k)*a(n-k-1) for n>0 with a(0)=1.
%F a(n) ~ c * 2^((n-1)*(n+4)/2), where c = 0.71662215139236633556752111264619992099204134882... - _Vaclav Kotesovec_, Feb 22 2014
%e G.f.: A(x) = 1 + x + 6*x^2 + 91*x^3 + 2910*x^4 + 187178*x^5 + 24019884*x^6 +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 13*x^2 + 194*x^3 + 6038*x^4 + 381268*x^5 + 48457325*x^6 + 12358976074*x^7 + 6315716731394*x^8 + 6461044887240556*x^9 +...
%e such that a(n) = (2^n-1) times the coefficient of x^(n-1) in A(x)^2:
%e a(2) = 3 * 2 = 6;
%e a(3) = 7 * 13 = 91;
%e a(4) = 15 * 194 = 2910;
%e a(5) = 31 * 6038 = 187178;
%e a(6) = 63 * 381268 = 24019884; ...
%t a = ConstantArray[0,21]; a[[1]]=1; a[[2]]=1; Do[a[[n+1]] = (2^n-1)* Sum[a[[k+1]]*a[[n-k]],{k,0,n-1}],{n,2,20}]; a (* _Vaclav Kotesovec_, Feb 22 2014 *)
%o (PARI) /* Generating Function Satisfies: */
%o {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} /* = n-th derivative of F */
%o {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+sum(k=1, n, x^k/k!*Dx(k, x*A^2+x*O(x^n) ))); polcoeff(A, n)}
%o (PARI) /* Recurrence: */
%o {a(n)=if(n==0,1,(2^n-1)*sum(k=0,n-1,a(k)*a(n-k-1)))}
%o for(n=0,15,print1(a(n),", "))
%o (PARI) /* Recurrence: */
%o {a(n)=local(A=1+sum(k=1,n-1,a(k)*x^k)+x*O(x^n));if(n==0,1,(2^n-1)*polcoeff(A^2,n-1))}
%Y Cf. A005329, A182264.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 21 2012
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