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A193038 G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^sigma(n), where sigma(n) = sum of divisors of n (A000203).

%I A193038 #8 May 16 2025 17:35:08
%S A193038 1,1,2,6,23,101,475,2321,11629,59364,307648,1614724,8567810,45890927,
%T A193038 247817187,1347819147,7376472346,40594360200,224500075274,
%U A193038 1247028876157,6954322550810,38921347036195,218541728743211,1230754878156173,6950114772716368
%N A193038 G.f. A(x) satisfies: x = Sum_{n>=1} x^n*A(-x)^sigma(n), where sigma(n) = sum of divisors of n (A000203).
%C A193038 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).
%e A193038 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 101*x^5 + 475*x^6 +...
%e A193038 The g.f. satisfies:
%e A193038 x = x*A(-x) + x^2*A(-x)^3 + x^3*A(-x)^4 + x^4*A(-x)^7 + x^5*A(-x)^6 + x^6*A(-x)^12 +...+ x^n*A(-x)^A000203(n) +...
%e A193038 where A000203 begins: [1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,31,...].
%o A193038 (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)^sigma(m)),#A));if(n<0,0,A[n+1])}
%Y A193038 Cf. A193036, A193037, A193039, A193040, A000203.
%K A193038 nonn
%O A193038 0,3
%A A193038 _Paul D. Hanna_, Jul 14 2011