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

%I A380552 #13 Feb 17 2025 17:38:23
%S A380552 1,2,14,88,611,4372,32889,254384,2017341,16300550,133767542,
%T A380552 1111727456,9338434699,79155402978,676196048434,5815796615520,
%U A380552 50318860986107,437662918037250,3824609516638443,33563127916092808,295655735395364616,2613391671434553220,23173063762591336049,206066197523415007168
%N A380552 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ).
%C A380552 Moebius transform of A006632.
%H A380552 Paul D. Hanna, <a href="/A380552/b380552.txt">Table of n, a(n) for n = 1..500</a>
%F A380552 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A380552 (1) x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ).
%F A380552 (2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(3*n) / (1 - x^n*(1-x)^(3*n)).
%F A380552 (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.
%F A380552 (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.
%e A380552 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 + ...
%e A380552 where x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ).
%e A380552 RELATED SERIES.
%e A380552 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) + ...
%e A380552 which equals x*F(x)^3 where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
%o A380552 (PARI) \\ As the Moebius transform of A006632 \\
%o A380552 {a(n) = sumdiv(n,d, moebius(n/d) * binomial(4*d-1,d-1)*3/(4*d-1) )}
%o A380552 for(n=1,30,print1(a(n),", "))
%o A380552 (PARI) \\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(3*n) ) \\
%o A380552 {a(n) = my(V=[0,1]); for(i=0,n, V = concat(V,0); A = Ser(V);
%o A380552 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]}
%o A380552 for(n=1,30,print1(a(n),", "))
%Y A380552 Cf. A346935, A034742, A380551, A380553, A006632, A002293, A008683.
%K A380552 nonn
%O A380552 1,2
%A A380552 _Paul D. Hanna_, Feb 16 2025