OFFSET
0,3
FORMULA
G.f.: A(x) = Product_{n>0} 1/(1-x^n-x^(2*n))^a(n-1).
G.f.: A(x) = exp( Sum(n>=1} x^n/n * Sum_{k=0..n} C(n,k)*x^k*A(x^(n+k)) ). [From Paul D. Hanna, Nov 03 2011]
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 19*x^4 + 48*x^5 + 134*x^6 +...
where
log(A(x)) = A(x)*x + 3*A(x^2)*x^2/2 + 4*A(x^3)*x^3/3 + 7*A(x^4)*x^4/4 + 11*A(x^5)*x^5/5 +...
Equivalently,
log(A(x)) = (A(x)+x*A(x^2))*x + (A(x^2)+2*x*A(x^3)+x^2*A(x^4))*x^2/2 + (A(x^3)+3*x*A(x^4)+3*x^2*A(x^5)+x^3*A(x^6))*x^3/3 + (A(x^4)+4*x*A(x^5)+6*x^2*A(x^6)+4*x^3*A(x^7)+x^4*A(x^8))*x^4/4 +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m/m*sum(k=0, m, binomial(m, k)*x^k*subst(A, x, x^(m+k)+x*O(x^n)))))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 17 2002
STATUS
approved