%I #18 Feb 21 2018 03:04:17
%S 1,1,1,3,10,34,112,382,1352,4884,17856,66022,246764,930878,3538788,
%T 13542716,52133416,201746212,784378792,3062431132,12001867312,
%U 47197716460,186187480816,736582735738,2921679555340,11617001425938,46294191726972,184866924629832
%N G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^A006519(n), where A006519(n) is the highest power of 2 dividing n.
%C Compare the g.f. to a g.f. C(x) of the Catalan numbers: x = Sum_{n>=1} x^n*C(-x)^(2*n-1).
%H Paul D. Hanna, <a href="/A193036/b193036.txt">Table of n, a(n) for n = 0..720</a>
%F G.f. satisfies: x = Sum_{n>=0} x^(2^n) * A(-x)^(2^n) / (1 - x^(2*2^n)).
%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 34*x^5 + 112*x^6 + ...
%e The g.f. satisfies:
%e x = x*A(-x) + x^2*A(-x)^2 + x^3*A(-x) + x^4*A(-x)^4 + x^5*A(-x) + x^6*A(-x)^2 + x^7*A(-x) + x^8*A(-x)^8 + x^9*A(-x) + ... + x^n * A(-x)^A006519(n) + ...
%e where A006519 begins: [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,...].
%e The g.f. also satisfies:
%e x = x*A(-x)/(1-x^2) + x^2*A(-x)^2/(1-x^4) + x^4*A(-x)^4/(1-x^8) + x^8*A(-x)^8/(1-x^16) + x^16*A(-x)^16/(1-x^32) + x^32*A(-x)^32/(1-x^64) + ...
%e Related table.
%e The table of coefficients in A(-x)^(2^n) / (1 - x^(2*2^n)) begins:
%e n=0: [1, -1, 2, -4, 12, -38, 124, -420, 1476, -5304, ...];
%e n=1: [1, -2, 3, -8, 28, -96, 324, -1124, 4024, -14684, ...];
%e n=2: [1, -4, 10, -28, 95, -344, 1244, -4512, 16616, -62072, ...];
%e n=3: [1, -8, 36, -136, 514, -2008, 7924, -31176, 122495, ...];
%e n=4: [1, -16, 136, -848, 4500, -22032, 103480, -473520, ...];
%e n=5: [1, -32, 528, -6048, 54632, -418720, 2855088, ...];
%e n=6: [1, -64, 2080, -45888, 775120, -10720576, 126777952, ...];
%e n=7: [1, -128, 8256, -358016, 11750304, -311550592, 6955997376, ...];
%e ...
%e where x = Sum_{n>=0} x^(2^n) * A(-x)^(2^n) / (1 - x^(2*2^n)).
%o (PARI) {a(n)=local(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=polcoeff(sum(m=1,#A,(-x)^m*Ser(A)^(2^valuation(m,2))),#A));if(n<0,0,A[n+1])}
%Y Cf. A193037, A193038, A193039, A193040, A006519.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jul 14 2011