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A193040 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A131507(n), where A131507 is defined as "2*n+1 appears n times.".

%I A193040 #17 Mar 30 2012 18:37:27
%S A193040 1,1,2,7,29,129,600,2889,14293,72228,371208,1934236,10194853,54258010,
%T A193040 291175463,1573878211,8560931357,46825444031,257386132988,
%U A193040 1421034475176,7876770462043,43817869686744,244552276036950,1368945007588648,7683977372121530
%N A193040 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A131507(n), where A131507 is defined as "2*n+1 appears n times.".
%C A193040 Compare the g.f. to a g.f. C(x) of the Catalan numbers: 1 = Sum_{n>=0} x^n*C(-x)^(2*n+1).
%F A193040 G.f. satisfies: 1-x = Sum_{n>=1} x^(n*(n-1)/2)* (1-x^n)* A(-x)^(2*n-1).
%e A193040 G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 129*x^5 + 600*x^6 +...
%e A193040 The g.f. satisfies:
%e A193040 1 = A(-x) + x*A(-x)^3 + x^2*A(-x)^3 + x^3*A(-x)^5 + x^4*A(-x)^5 + x^5*A(-x)^5 + x^6*A(-x)^7 +...+ x^n*A(-x)^A131507(n) +...
%e A193040 where A131507 begins: [1,3,3,5,5,5,7,7,7,7,9,9,9,9,9,11,...].
%e A193040 The g.f. also satisfies:
%e A193040 1-x = (1-x)*A(-x) + x*(1-x^2)*A(-x)^3 + x^3*(1-x^3)*A(-x)^5 + x^6*(1-x^4)*A(-x)^7 + x^10*(1-x^5)*A(-x)^9 +...
%o A193040 (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*floor(sqrt(2*m)+1/2)-1) ),#A));if(n<0,0,A[n+1])}
%Y A193040 Cf. A193039, A193037, A192455, A193050, A131507.
%K A193040 nonn
%O A193040 0,3
%A A193040 _Paul D. Hanna_, Jul 14 2011