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FORMULA
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Given g.f. A(x), the following sums are all equal.
(1) B(x) = Sum_{n>=0} x^n*A(x)^(1*n) / (1 - x*A(x)^(1*n+4));
(2) B(x) = Sum_{n>=0} x^n*A(x)^(2*n) / (1 - x*A(x)^(3*n+3));
(3) B(x) = Sum_{n>=0} x^n*A(x)^(3*n) / (1 - x*A(x)^(3*n+2));
(4) B(x) = Sum_{n>=0} x^n*A(x)^(4*n) / (1 - x*A(x)^(1*n+1));
(5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+5*n) * (1 - x^2*A(x)^(2*n+5)) / ((1 - x*A(x)^(n+1))*(1 - x*A(x)^(n+4))),
(6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(3*n^2+5*n) * (1 - x^2*A(x)^(6*n+5)) / ((1 - x*A(x)^(3*n+2))*(1 - x*A(x)^(3*n+3)));
see the example section for the value of B(x).
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EXAMPLE
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G.f.: A(x) = 1 + x + 5*x^2 + 43*x^3 + 443*x^4 + 5009*x^5 + 60104*x^6 + 751778*x^7 + 9696036*x^8 + 128037209*x^9 + 1722632206*x^10 + ...
such that the following sums are all equal:
(1) B(x) = 1/(1 - x*A(x)^4) + x*A(x)^1/(1 - x*A(x)^5) + x^2*A(x)^2/(1 - x*A(x)^6) + x^3*A(x)^3/(1 - x*A(x)^7) + x^4*A(x)^4/(1 - x*A(x)^8) + ...
(2) B(x) = 1/(1 - x*A(x)^3) + x*A(x)^2/(1 - x*A(x)^6) + x^2*A(x)^4/(1 - x*A(x)^9) + x^3*A(x)^6/(1 - x*A(x)^12) + x^4*A(x)^8/(1 - x*A(x)^15) + ...
(3) B(x) = 1/(1 - x*A(x)^2) + x*A(x)^3/(1 - x*A(x)^5) + x^2*A(x)^6/(1 - x*A(x)^8) + x^3*A(x)^9/(1 - x*A(x)^11) + x^4*A(x)^12/(1 - x*A(x)^14) + ...
(4) B(x) = 1/(1 - x*A(x)^1) + x*A(x)^4/(1 - x*A(x)^2) + x^2*A(x)^8/(1 - x*A(x)^3) + x^3*A(x)^12/(1 - x*A(x)^4) + x^4*A(x)^16/(1 - x*A(x)^5) + ...
where
B(x) = 1 + 2*x + 8*x^2 + 51*x^3 + 442*x^4 + 4534*x^5 + 51182*x^6 + 613806*x^7 + 7675397*x^8 + 98971497*x^9 + 1306630823*x^10 + 17575262387*x^11 + 240012293969*x^12 + ... (see A351772).
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PROG
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(PARI) {a(n) = my(A=[1, 1, 0]); for(i=0, n, A=concat(A, 0);
B1 = sum(m=0, #A, x^m*Ser(A)^(2*m)/(1 - x*Ser(A)^(3*m+3)) );
B2 = sum(m=0, #A, x^m*Ser(A)^(4*m)/(1 - x*Ser(A)^(1*m+1)) );
A[#A-1] = polcoeff((B1 - B2)/2, #A); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1, 1, 0]); for(i=0, n, A=concat(A, 0);
B1 = sum(m=0, #A, x^m*Ser(A)^(3*m)/(1 - x*Ser(A)^(3*m+2)) );
B2 = sum(m=0, #A, x^m*Ser(A)^(1*m)/(1 - x*Ser(A)^(1*m+4)) );
A[#A-1] = polcoeff((B1 - B2)/2, #A); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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