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G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)*A(x^(n+k)) ).
1

%I #9 Mar 30 2012 18:37:32

%S 1,2,6,20,63,213,719,2505,8864,31948,116725,432074,1616022,6100775,

%T 23214144,88949045,342899474,1329020016,5175758820,20243197030,

%U 79480515302,313155710230,1237771800135,4906616164195,19502048947616,77703941363599,310305199056779

%N G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)*A(x^(n+k)) ).

%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 63*x^4 + 213*x^5 + 719*x^6 +...

%e where

%e log(A(x)) = (A(x)+A(x^2))*x + (A(x^2)+2*A(x^3)+A(x^4))*x^2/2 + (A(x^3)+3*A(x^4)+3*A(x^5)+A(x^6))*x^3/3 + (A(x^4)+4*A(x^5)+6*A(x^6)+4*A(x^7)+A(x^8))*x^4/4 +...

%e Explicitly,

%e log(A(x)) = 2*x + 8*x^2/2 + 32*x^3/3 + 100*x^4/4 + 387*x^5/5 + 1370*x^6/6 + 5315*x^7/7 + 20444*x^8/8 + 80897*x^9/9 + 320883*x^10/10 +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m,k)*subst(A,x,x^(m+k)+x*O(x^n))))));polcoeff(A,n)}

%Y Cf. A073063.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 03 2011