OFFSET
0,3
FORMULA
Define f(z,q) = Sum_{n>=0} z^n * q^(n^2) then g.f. A(q) satisfies:
A(q) = (f(A(q),q) - q*f(q^2*A(q),q))/(1-q).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 17*x^5 + 37*x^6 + 82*x^7 +...
where the g.f. A(x) satisfies the equivalent expressions:
A(x) = 1 + A(x)*x*(1-x^3)/(1-x) + A(x)^2*x^4*(1-x^5)/(1-x) + A(x)^3*x^9*(1-x^7)/(1-x) + A(x)^4*x^16*(1-x^9)/(1-x) +...
A(x) = 1 + A(x)*(x + x^2 + x^3) + A(x)^2*(x^4 + x^5 + x^6 + x^7 + x^8) + A(x)^3*(x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15) +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, sqrtint(n+1), A^m*x^(m^2)*(1-x^(2*m+1))/(1-x))+x*O(x^n)); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 06 2011
STATUS
approved