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O.g.f. A(x) satisfies: A(x) = 1 + Integral (x*A(x)^4)' / (x*A(x))' dx.
4

%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