OFFSET
0,4
COMMENTS
Compare to: exp( Sum_{n>=1} x^(2*n-1)/(2*n-1) ) = sqrt(1-x^2)/(1-x).
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 75*x^5/5! +...
where
log(A(x)) = x + (1+1/2)*x^3/3 + (1+1/2+1/3)*x^5/5 + (1+1/2+1/3+1/4)*x^7/7 + (1+1/2+1/3+1/4+1/5)*x^9/9 + (1+1/2+1/3+1/4+1/5+1/6)*x^11/11 +...
Explicitly,
log(A(x)) = x + 1/2*x^3 + 11/30*x^5 + 25/84*x^7 + 137/540*x^9 + 49/220*x^11 + 363/1820*x^13 + 761/4200*x^15 +...
Equivalently,
log(A(x)) = x + 3*x^3/3! + 44*x^5/5! + 1500*x^7/7! + 92064*x^9/9! + 8890560*x^11/11! + 1241982720*x^13/13! + 236938141440*x^15/15! +...
PROG
(PARI) {H(n)=sum(k=1, n, 1/k)}
{a(n)=local(A=1); A=exp(sum(k=1, n\2+1, H(k)*x^(2*k-1)/(2*k-1))+x*O(x^n)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 08 2014
STATUS
approved