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FORMULA
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The g.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} (2*n+1)*x^n*A(x)^(2*n)/(1 - x^(2*n+1)*A(x)^(4*n+2)),
(2) A(x)^2 = Sum_{n>=0} (n+1)^3*x^n*A(x)^(2*n)/(1 - x^(2*n+2)*A(x)^(4*n+4)),
(3) A(x) = Product_{n>=1} (1 + x^n*A(x)^(2*n))^4*(1 - x^(2*n)*A(x)^(4*n))^4,
(4) A(x) = exp( Sum_{n>=1} 4*(x^n*A(x)^(2*n)/(1 + x^n*A(x)^(2*n)))/n ),
(5) A(x/F(x)^8) = F(x)^4 where F(x) = Sum_{n>=0} x^(n*(n+1)/2),
due to q-series identities.
Self-convolution 2nd power equals A194042.
Self-convolution 4th root equals A194043.
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EXAMPLE
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G.f.: A(x) = 1 + 4*x + 38*x^2 + 472*x^3 + 6685*x^4 + 102340*x^5 +...
where
(0) A(x)^(1/4) = 1 + x*A(x)^2 + x^3*A(x)^6 + x^6*A(x)^12 + x^10*A(x)^20 + x^15*A(x)^30 + x^21*A(x)^42 +... +...
(1) A(x) = 1/(1-x*A(x)^2) + 3*x*A(x)^2/(1-x^3*A(x)^6) + 5*x^2*A(x)^4/(1-x^5*A(x)^10) + 7*x^3*A(x)^6/(1-x^7*A(x)^14) +...
(2) A(x)^2 = 1/(1-x^2*A(x)^4) + 8*x*A(x)^2/(1-x^4*A(x)^8) + 27*x^2*A(x)^4/(1-x^6*A(x)^12) + 64*x^3*A(x)^6/(1-x^8*A(x)^16) +...
(3) A(x) = (1+x*A(x)^2)^4*(1-x^2*A(x)^4)^4 * (1+x^2*A(x)^4)^4*(1-x^4*A(x)^8)^4 * (1+x^3*A(x)^6)^4*(1-x^6*A(x)^12)^4 * (1+x^4*A(x)^8)^4*(1-x^8*A(x)^16)^4 *...
(4) log(A(x)) = 4*x*A(x)^2/(1+x*A(x)^2) + 4*(x^2*A(x)^4/(1+x^2*A(x)^4))/2 + 4*(x^3*A(x)^6/(1+x^3*A(x)^6))/3 + 4*(x^4*A(x)^8/(1+x^4*A(x)^8))/4 +...
Related expansions begin:
_ A(x)^(1/4) = 1 + x + 8*x^2 + 93*x^3 + 1272*x^4 + 19058*x^5 + 302705*x^6 + 5007234*x^7 + 85341048*x^8 +...+ A194043(n)*x^n +...
_ A(x)^2 = 1 + 8*x + 92*x^2 + 1248*x^3 + 18590*x^4 + 294032*x^5 + 4848456*x^6 + 82433472*x^7 + 1434755717*x^8 +...+ A194042(n)*x^n +...
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PROG
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(PARI) {a(n)=local(A=1+x, T=sum(m=0, sqrtint(2*n+1), x^(m*(m+1)/2))+x*O(x^n)); A=(serreverse(x/T^8)/x)^(1/2); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (x*A^2+x*O(x^n))^(m*(m+1)/2))^4); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (2*m+1)*(x*A^2)^m/(1-(x*A^2+x*O(x^n))^(2*m+1)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, (m+1)^3*(x*A^2)^m/(1-(x*A^2+x*O(x^n))^(2*m+2)))^(1/2)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+(x*A^2)^m)*(1-(x*A^2)^(2*m)+x*O(x^n)))^4); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, 4*(x*A^2)^m/(1+(x*A^2)^m+x*O(x^n))/m))); polcoeff(A, n)}
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