OFFSET
0,3
COMMENTS
Compare to: G(x) = 1 + Integral G(2*x)^(1/2) dx holds when G(x) = exp(x).
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 12*x^3/3! + 312*x^4/4! + 37008*x^5/5! + 18540576*x^6/6! + 37740977856*x^7/7! + 308640640553856*x^8/8! + 10108585206574665984*x^9/9! + ...
Related series.
A(4*x)^(1/2) = 1 + 2*x + 12*x^2/2! + 312*x^3/3! + 37008*x^4/4! + 18540576*x^5/5! + 37740977856*x^6/6! + ... + a(n+1)*x^n/n! + ...
A(2*x)^(1/2) = 1 + x + 3*x^2/2! + 39*x^3/3! + 2313*x^4/4! + 579393*x^5/5! + 589702779*x^6/6! + ... + A300046(n)*x^n/n! + ...
log(A(x)) = x + x^2/2! + 8*x^3/3! + 270*x^4/4! + 35472*x^5/5! + 18318288*x^6/6! + ... + A300047(n)*x^n/n! + ...
PROG
(PARI) {a(n) = my(A=1+x); for(i=1, n, A = 1 + intformal(subst(A, x, 4*x)^(1/2) +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 16, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 25 2018
STATUS
approved