OFFSET
0,3
FORMULA
G.f. satisfies the identities:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(n+1) / Product_{k=1..n} (1 - x^k*A(x)).
(2) A(x) = 1/(1 - Sum_{n>=1} x^n*A(x)^n / Product_{k=1..n} (1 - x^k*A(x)) ).
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 91*x^4 + 489*x^5 + 2751*x^6 +...
where the g.f. satisfies:
(0) A(x) = 1 + x*A(x)^2/(1-x*A(x))^2 + x^4*A(x)^4/((1-x*A(x))^2*(1-x^2*A(x))^2) + x^9*A(x)^6/((1-x*A(x))^2*(1-x^2*A(x))^2*(1-x^3*A(x))^2) +...
(1) A(x) = 1 + x*A(x)^2/(1-x*A(x)) + x^2*A(x)^3/((1-x*A(x))*(1-x^2*A(x))) + x^3*A(x)^4/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^(2*m)/prod(k=1, m, 1-x^k*A+x*O(x^n))^2)); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^(m+1)/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 11 2012
STATUS
approved