OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 + x = Product_{n>=1} (1 - (-x)^n*A(x)).
(2) 1 + x = Sum_{n>=0} (-1)^n * (-x)^(n*(n+1)/2) * A(x)^n / (Product_{k=1..n} (1 - (-x)^k)).
(3) 1/(1+x) = Sum_{n>=0} (-x)^n * A(x)^n / Product_{k=1..n} (1 - (-x)^k).
(4) log(1+x) = Sum_{n>=1} -(-x)^n * Sum_{d|n} A(x)^(n/d) * d/n.
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 20*x^6 + 45*x^7 + 102*x^8 + 235*x^9 + 550*x^10 + 1304*x^11 + 3121*x^12 + ...
where A(x) solves the infinite product equation
(1) 1 + x = (1 + x*A(x)) * (1 - x^2*A(x)) * (1 + x^3*A(x)) * (1 - x^4*A(x)) * (1 + x^5*A(x)) * (1 - x^6*A(x)) * ...
as well as the following infinite series equations
(2) 1 + x = 1 + x*A(x)/(1+x) - x^3*A(x)^2/((1+x)*(1-x^2)) - x^6*A(x)^3/((1+x)*(1-x^2)*(1+x^3)) + x^10*A(x)^4/((1+x)*(1-x^2)*(1+x^3)*(1-x^4)) + x^15*A(x)^5/((1+x)*(1-x^2)*(1+x^3)*(1-x^4)*(1+x^5)) --++ ...
(3) 1/(1+x) = 1 - x*A(x)/(1+x) + x^2*A(x)^2/((1+x)*(1-x^2)) - x^3*A(x)^3/((1+x)*(1-x^2)*(1+x^3)) + x^4*A(x)^4/((1+x)*(1-x^2)*(1+x^3)*(1-x^4)) - x^5*A(x)^5/((1+x)*(1-x^2)*(1+x^3)*(1-x^4)*(1+x^5)) +- ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = -polcoeff( prod(n=1, #A, 1 - (-x)^n*Ser(A) ), #A) ); A[n+1]}
for(n=0, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 28 2022
STATUS
approved