%I #24 Apr 20 2018 00:01:18
%S 1,1,3,11,43,171,677,2637,10035,37171,134009,472785,1655845,5910373,
%T 22254507,90625475,396822579,1803795507,8151776201,35314777505,
%U 142395796689,518352934225,1625522953935,3944383216263,4604242439037,-17114536692099,-114353748666873,-52384917067153,4112292989447275,42810794269242411,290607272326013813
%N O.g.f. A(x) satisfies: A(x) = 1 + Integral (x*A(x)^4)' / (x*A(x))' dx.
%C Note: if F(x) = 1 + Integral (x*F(x)^3)' / (x*F(x))' dx, then F(x) does not consist entirely of integer coefficients in its power series expansion.
%C Given G(x) = 1 + Integral (x*G(x)^p)' / (x*G(x)^q)' dx, for what fixed integers p and q is G(x) an integer power series?
%H Paul D. Hanna, <a href="/A302705/b302705.txt">Table of n, a(n) for n = 0..520</a>
%F O.g.f. A(x) satisfies:
%F (1) A(x) = 1 + Integral (x*A(x)^4)' / (x*A(x))' dx.
%F (2) A(x) = 1 + Integral A(x)^3 * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)) dx.
%F (3) A(x) = 1 + Integral A(x) * ( sqrt(1 - 4*x*A(x)^2 + 16*x^2*A(x)^4) - 1 + 4*x*A(x)^2 ) / (2*x) dx.
%F (4) 0 = A(x)^4 - A(x)*(1 - 4*x*A(x)^2)*A'(x) - x*A'(x)^2.
%e O.g.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 171*x^5 + 677*x^6 + 2637*x^7 + 10035*x^8 + 37171*x^9 + 134009*x^10 + 472785*x^11 + 1655845*x^12 + ...
%e RELATED SERIES.
%e (x*A(x)^4)' / (x*A(x))' = 1 + 6*x + 33*x^2 + 172*x^3 + 855*x^4 + 4062*x^5 + 18459*x^6 + 80280*x^7 + 334539*x^8 + ... + (n+1)*a(n+1)*x^n + ...
%e which equals A'(x).
%e The logarithmic derivative of the g.f. begins:
%e A'(x)/A(x) = 1 + 5*x + 25*x^2 + 121*x^3 + 561*x^4 + 2477*x^5 + 10361*x^6 + 40817*x^7 + 150433*x^8 + 515605*x^9 + 1646041*x^10 + ...
%e which equals (sqrt(1 - 4*x*A(x)^2 + 16*x^2*A(x)^4) - 1 + 4*x*A(x)^2) / (2*x).
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = A = 1 + intformal( (x*A^4)'/(x*A +x*O(x^n))' );); polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A302701, A302704, A303064.
%K sign
%O 0,3
%A _Paul D. Hanna_, Apr 15 2018