A380551 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ).

1, 1, 6, 28, 142, 720, 3875, 21288, 120168, 690546, 4032014, 23840724, 142498691, 859512043, 5225263875, 31983651216, 196947587822, 1219199232294, 7583142491924, 47365473951152, 296983176365613, 1868545308601424, 11793499763070479, 74650344221104632, 473770694965305205, 3014124873709172435

Offset: 1

Author

Paul D. Hanna, Feb 16 2025

Keywords

• nonn

Comments

Moebius transform of A006013.

Examples

			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.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A346925, A034742, A380552, A380553, A006013, A001764, A008683.

Programs

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),", "))

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.