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A162661 G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x)^2 ). 0

%I

%S 1,1,2,7,33,189,1249,9237,74972,659042,6215154,62435805,664459091,

%T 7458334388,87979090059,1087309348481,14041705640439,189050930463638,

%U 2648140182064473,38521885088392896,580970615943277573

%N G.f. satisfies: A(x) = 1 + x*A(x) * A( x*A(x)^2 ).

%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n with a(0,m)=1, then

%F a(n,m) = Sum_{k=0..n} m*C(2n-k+m,k)/(2n-k+m) * a(n-k,k).

%F ...

%F Also, if log(A(x)) = Sum_{n>=0} L(n)*x^n/n, then

%F L(n) = n*Sum_{k=1..n} C(2n-k,k)/(2n-k) * a(n-k,k).

%F ...

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 33*x^4 + 189*x^5 +...

%e A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 84*x^4 + 472*x^5 +...

%e A(x*A(x)^2) = 1 + x + 4*x^2 + 20*x^3 + 121*x^4 + 838*x^5 +...

%e log(A(x)) = x + 3/2*x^2 + 16/3*x^3 + 103/4*x^4 + 756/5*x^5 +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A*subst(A,x,x*A^2+O(x^n)));polcoeff(A,n)}

%o (PARI) {a(n,m=1)=if(n==0,1,if(m==0,0^n,sum(k=0,n,m*binomial(2*n-k+m,k)/(2*n-k+m)*a(n-k,k))))}

%o (PARI) /* log(A(x)) = Sum_{n>=0} L(n)*x^n/n where: */

%o {L(n)=if(n<1,0,n*sum(k=1,n,binomial(2*n-k,k)/(2*n-k)*a(n-k,k)))}

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 09 2009

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Last modified June 14 05:10 EDT 2021. Contains 345018 sequences. (Running on oeis4.)