A380552 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ).

1, 2, 14, 88, 611, 4372, 32889, 254384, 2017341, 16300550, 133767542, 1111727456, 9338434699, 79155402978, 676196048434, 5815796615520, 50318860986107, 437662918037250, 3824609516638443, 33563127916092808, 295655735395364616, 2613391671434553220, 23173063762591336049, 206066197523415007168

Offset

1,2

Comments

Moebius transform of A006632.

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)^(3*n) ).

(2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(3*n) / (1 - x^n*(1-x)^(3*n)).

(3) x*F(x)^3 = Sum_{n>=1} a(n) * x^n/(1-x^n) where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.

(4) a(n) = Sum_{d|n} mu(n/d) * binomial(4*d-1,d-1)*3/(4*d-1), where mu is the Moebius function A008683.

Example

G.f.: A(x) = x + 2*x^2 + 14*x^3 + 88*x^4 + 611*x^5 + 4372*x^6 + 32889*x^7 + 254384*x^8 + 2017341*x^9 + 16300550*x^10 + ...
where x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ).
RELATED SERIES.
Sum_{n>=1} a(n) * x^n/(1-x^n) = x + 3*x^2 + 15*x^3 + 91*x^4 + 612*x^5 + 4389*x^6 + 32890*x^7 + 254475*x^8 + ... + A006632(n)*x^(n) + ...
which equals x*F(x)^3 where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.

Prog

(PARI) \\ As the Moebius transform of A006632 \\
{a(n) = sumdiv(n, d, moebius(n/d) * binomial(4*d-1, d-1)*3/(4*d-1) )}
for(n=1, 30, print1(a(n), ", "))
(PARI) \\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(3*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)^(3*m) +x*O(x^#V)) ), #V-1)); V[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A346935, A034742, A380551, A380553, A006632, A002293, A008683.

Sequence in context: A065355 A162478 A348615 * A189392 A235374 A065892

Adjacent sequences: A380549 A380550 A380551 * A380553 A380554 A380555

Keyword

nonn,new

Author

Paul D. Hanna, Feb 16 2025

Status

approved