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%I #11 Mar 30 2012 18:37:26
%S 1,1,5,77,3191,332481,83495679,49089025473,66142622730623,
%T 200954040909283841,1359156203343916471295,20253823024219712679748609,
%U 659335186924510858208484730879,46554554840488755704034417937268737
%N G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1 + 2^n*x)^(2n+1).
%C G.f. satisfies a variant of an identity of the Catalan numbers (A000108):
%C 1 = Sum_{n>=0} A000108(n)*x^n/(1 + x)^(2n+1).
%C Also, g.f. satisfies a variant of an identity involving A003024:
%C 1 = Sum_{n>=0} A003024(n)*x^n/(1 + 2^n*x)^(n+1),
%C where A003024(n) is the number of acyclic digraphs with n labeled nodes.
%e G.f.: 1 = 1/(1+x) + x/(1+2*x)^3 + 5*x^2/(1+4*x)^5 + 77*x^3/(1+8*x)^7 + 3191*x^4/(1+16*x)^9 + 332481*x^5/(1+32*x)^11 +...
%o (PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+2^k*x+x*O(x^n))^(2*k+1)), n)}
%Y Cf. A188456, A188457, A003024.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 31 2011
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