A380553 - OEIS

%I #12 Feb 17 2025 17:38:36

%S 1,3,25,200,1770,16351,158223,1577328,16112031,167708890,1772645419,

%T 18974340640,205263418940,2240623110285,24648785800540,

%U 272994642782048,3041495503591364,34064252952038769,383302465665133013,4331178750570145160,49126274119206904221,559128033687856289017

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

%C Moebius transform of A118971.

%H Paul D. Hanna, <a href="/A380553/b380553.txt">Table of n, a(n) for n = 1..500</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).

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

%F (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.

%F (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.

%e 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 + ...

%e where x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).

%e RELATED SERIES.

%e 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) + ...

%e which equals x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.

%o (PARI) \\ As the Moebius transform of A118971 \\

%o {a(n) = sumdiv(n,d, moebius(n/d) * binomial(5*d-1,d-1)*4/(5*d-1) )}

%o for(n=1,30,print1(a(n),", "))

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

%o {a(n) = my(V=[0,1]); for(i=0,n, V = concat(V,0); A = Ser(V);

%o 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]}

%o for(n=1,30,print1(a(n),", "))

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

%K nonn,new

%O 1,2

%A _Paul D. Hanna_, Feb 16 2025