OFFSET
0,3
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 61*x^4 + 306*x^5 + 1623*x^6 +...
where the logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A)*x*A + (1 + 2^3*x*A + x^2*A^2)*x^2*A^2/2 +
(1 + 3^3*x*A + 3^3*x^2*A^2 + x^3*A^3)*x^3*A^3/3 +
(1 + 4^3*x*A + 6^3*x^2*A^2 + 4^3*x^3*A^3 + x^4*A^4)*x^4*A^4/4 +
(1 + 5^3*x*A + 10^3*x^2*A^2 + 10^3*x^3*A^3 + 5^3*x^4*A^4 + x^5*A^5)*x^5*A^5/5 +
(1 + 6^3*x*A + 15^3*x^2*A^2 + 20^3*x^3*A^3 + 15^3*x^4*A^4 + 6^3*x^5*A^5 + x^6*A^6)*x^6*A^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 31*x^3/3 + 185*x^4/4 + 1126*x^5/5 + 7043*x^6/6 + 44689*x^7/7 + 286241*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^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 14 2011
STATUS
approved