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A193050 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A003059(n+1), where A003059 is defined by "n appears 2n-1 times.".

%I A193050 #10 Jul 22 2025 12:19:00
%S A193050 1,1,1,2,4,8,17,38,87,204,489,1191,2938,7328,18448,46809,119583,
%T A193050 307324,793965,2060770,5371156,14051901,36887289,97131351,256488187,
%U A193050 679046184,1802047427,4792800096,12773166908,34106055493,91228795961,244427136822,655900969465
%N A193050 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A003059(n+1), where A003059 is defined by "n appears 2n-1 times.".
%C A193050 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 A193050 G.f. satisfies: 1-x = Sum_{n>=1} x^(n^2) * (1 - x^(2*n-1)) * A(-x)^n.
%e A193050 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 38*x^7 +...
%e A193050 The g.f. satisfies:
%e A193050 1 = A(-x) + x*A(-x)^2 + x^2*A(-x)^2 + x^3*A(-x)^2 + x^4*A(-x)^3 + x^5*A(-x)^3 + x^6*A(-x)^3 + x^7*A(-x)^3 + x^8*A(-x)^3 + x^9*A(-x)^4 +...+ x^n*A(-x)^A003059(n+1) +...
%e A193050 where A003059 begins: [1, 2,2,2, 3,3,3,3,3, 4,4,4,4,4,4,4, 5,...].
%e A193050 The g.f. also satisfies:
%e A193050 1-x = (1-x)*A(-x) + x*(1-x^3)*A(-x)^2 + x^4*(1-x^5)*A(-x)^3 + x^9*(1-x^7)*A(-x)^4 + x^16*(1-x^9)*A(-x)^5 +...
%o A193050 (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)^(1+sqrtint(m-1)) ), #A)); if(n<0, 0, A[n+1])}
%Y A193050 Cf. A192455, A193039, A193040, A003059.
%K A193050 nonn
%O A193050 0,4
%A A193050 _Paul D. Hanna_, Jul 14 2011