OFFSET
1,3
COMMENTS
Moebius transform of A006013.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..500
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ).
(2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(2*n) / (1 - x^n*(1-x)^(2*n)).
(3) x*F(x)^2 = Sum_{n>=1} a(n) * x^n/(1-x^n) where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
(4) a(n) = Sum_{d|n} mu(n/d) * binomial(3*d-1,d-1)*2/(3*d-1), where mu is the Moebius function A008683.
EXAMPLE
G.f.: A(x) = x + x^2 + 6*x^3 + 28*x^4 + 142*x^5 + 720*x^6 + 3875*x^7 + 21288*x^8 + 120168*x^9 + 690546*x^10 + ...
where x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ).
RELATED SERIES.
Sum_{n>=1} a(n) * x^n/(1-x^n) = x + 2*x^2 + 7*x^3 + 30*x^4 + 143*x^5 + 728*x^6 + 3876*x^7 + 21318*x^8 + ... + A006013(n)*x^(n+1) + ...
which equals x*F(x)^2 where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
PROG
(PARI) \\ As the Moebius transform of A006013 \\
{a(n) = sumdiv(n, d, moebius(n/d) * binomial(3*d-1, d-1)*2/(3*d-1) )}
for(n=1, 30, print1(a(n), ", "))
(PARI) \\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ) \\
{a(n) = my(V=[0, 1]); for(i=0, n, V = concat(V, 0); A = Ser(V);
V[#V] = polcoef(x - sum(m=1, #V, subst(A, x, x^m*(1-x)^(2*m) +x*O(x^#V)) ), #V-1)); V[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 16 2025
STATUS
approved