OFFSET
0,3
COMMENTS
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.
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?
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..520
FORMULA
O.g.f. A(x) satisfies:
(1) A(x) = 1 + Integral (x*A(x)^4)' / (x*A(x))' dx.
(2) A(x) = 1 + Integral A(x)^3 * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)) dx.
(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.
(4) 0 = A(x)^4 - A(x)*(1 - 4*x*A(x)^2)*A'(x) - x*A'(x)^2.
EXAMPLE
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 + ...
RELATED SERIES.
(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 + ...
which equals A'(x).
The logarithmic derivative of the g.f. begins:
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 + ...
which equals (sqrt(1 - 4*x*A(x)^2 + 16*x^2*A(x)^4) - 1 + 4*x*A(x)^2) / (2*x).
MAPLE
ogf := RootOf(16*x^2*(4*x-1)*y^3-8*x*(3*x-1)*y^2+(9*x+1)*(3*x-1)*y+1, y)^(1/2);
gfun[seriestolist](series(ogf, x=0, 31))[]; # Mark van Hoeij, Nov 28 2024
PROG
(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)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Apr 15 2018
STATUS
approved