OFFSET
0,3
COMMENTS
Compare to g.f. G(x) = (1+x^2)/(1-x-x^3) that satisfies:
G(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k / G(x)^k] * x^n/n ).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 22*x^5 + 58*x^6 +...
where
log(A(x)) = (1 + x/A(x))*x + (1 + 2^3*x/A(x) + x^2/A(x)^2)*x^2/2 +
(1 + 3^3*x/A(x) + 3^3*x^2/A(x)^2 + x^3/A(x)^3)*x^3/3 +
(1 + 4^3*x/A(x) + 6^3*x^2/A(x)^2 + 4^3*x^3/A(x)^3 + x^4/A(x)^4)*x^4/4 +
(1 + 5^3*x/A(x) + 10^3*x^2/A(x)^2 + 10^3*x^3/A(x)^3 + 5^3*x^4/A(x)^4 + x^5/A(x)^5)*x^5/5 +...
more explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 19*x^4/4 + 46*x^5/5 + 162*x^6/6 + 477*x^7/7 + 1371*x^8/8 +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^3*x^j/(A+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 31 2011
STATUS
approved