OFFSET
0,3
COMMENTS
If g.f. A(x) satisfies: A(x) = 1 + Integral (x/A(x)^p)' / (x/A(x)^q)' dx, then only for these pairs [p,q] are the coefficients of the series positive integers: [1,4], [4,7], [4,10], [5,9], [8,15], for 1 <= p <= 20, 1 <= q <= 20, p<>q.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..400
FORMULA
a(n) ~ 2^(n + 1/3) * 3^((3*n - 1)/2) * 5^(5*n/8 - 47/24) / (sqrt(Pi) * n^(5/2)).
MATHEMATICA
nmax = 25; A = 1; Do[A = 1 + Integrate[D[x/A^5, x]/D[x/A^9, x], x] + O[x]^nmax, nmax]; CoefficientList[A, x]
PROG
(PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + intformal( (x/A^5)'/(x/A^9 +x*O(x^n))' ); ); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Oct 15 2020
STATUS
approved