OFFSET
0,4
COMMENTS
Compare the g.f. to a g.f. C(x) of the Catalan numbers: 1 = Sum_{n>=0} x^n*C(-x)^(2*n+1).
FORMULA
G.f. satisfies: 1-x = Sum_{n>=1} x^(n^2) * (1 - x^(2*n-1)) * A(-x)^n.
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 38*x^7 +...
The g.f. satisfies:
1 = A(-x) + x*A(-x)^2 + x^2*A(-x)^2 + x^3*A(-x)^2 + x^4*A(-x)^3 + x^5*A(-x)^3 + x^6*A(-x)^3 + x^7*A(-x)^3 + x^8*A(-x)^3 + x^9*A(-x)^4 +...+ x^n*A(-x)^A003059(n+1) +...
where A003059 begins: [1, 2,2,2, 3,3,3,3,3, 4,4,4,4,4,4,4, 5,...].
The g.f. also satisfies:
1-x = (1-x)*A(-x) + x*(1-x^3)*A(-x)^2 + x^4*(1-x^5)*A(-x)^3 + x^9*(1-x^7)*A(-x)^4 + x^16*(1-x^9)*A(-x)^5 +...
PROG
(PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(sum(m=1, #A, (-x)^m*Ser(A)^(1+sqrtint(m-1)) ), #A)); if(n<0, 0, A[n+1])}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 14 2011
STATUS
approved