A380553 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).

1, 3, 25, 200, 1770, 16351, 158223, 1577328, 16112031, 167708890, 1772645419, 18974340640, 205263418940, 2240623110285, 24648785800540, 272994642782048, 3041495503591364, 34064252952038769, 383302465665133013, 4331178750570145160, 49126274119206904221, 559128033687856289017

Offset: 1

Author

Paul D. Hanna, Feb 16 2025

Keywords

• nonn

Comments

Moebius transform of A118971.

Examples

			G.f.: A(x) = x + 3*x^2 + 25*x^3 + 200*x^4 + 1770*x^5 + 16351*x^6 + 158223*x^7 + 1577328*x^8 + 16112031*x^9 + 167708890*x^10 + ...
where x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
RELATED SERIES.
Sum_{n>=1} a(n) * x^n/(1-x^n) = x + 4*x^2 + 26*x^3 + 204*x^4 + 1771*x^5 + 16380*x^6 + 158224*x^7 + 1577532*x^8 + ... + A118971(n)*x^(n) + ...
which equals x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.
		

Links

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

Crossrefs

Cf. A346936, A034742, A380551, A380552, A118971, A002294, A008683.

Programs

PARI

\\ As the Moebius transform of A118971 \\
{a(n) = sumdiv(n,d, moebius(n/d) * binomial(5*d-1,d-1)*4/(5*d-1) )}
for(n=1,30,print1(a(n),", "))

PARI

\\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(4*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)^(4*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)^(4*n) ).

(2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(4*n) / (1 - x^n*(1-x)^(4*n)).

(3) x*F(x)^4 = Sum_{n>=1} a(n) * x^n/(1-x^n) where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.

(4) a(n) = Sum_{d|n} mu(n/d) * binomial(5*d-1,d-1)*4/(5*d-1), where mu is the Moebius function A008683.