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

%I A380553 #12 Feb 17 2025 17:38:36
%S A380553 1,3,25,200,1770,16351,158223,1577328,16112031,167708890,1772645419,
%T A380553 18974340640,205263418940,2240623110285,24648785800540,
%U A380553 272994642782048,3041495503591364,34064252952038769,383302465665133013,4331178750570145160,49126274119206904221,559128033687856289017
%N A380553 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
%C A380553 Moebius transform of A118971.
%H A380553 Paul D. Hanna, <a href="/A380553/b380553.txt">Table of n, a(n) for n = 1..500</a>
%F A380553 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A380553 (1) x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
%F A380553 (2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(4*n) / (1 - x^n*(1-x)^(4*n)).
%F A380553 (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 A380553 (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 A380553 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 A380553 where x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ).
%e A380553 RELATED SERIES.
%e A380553 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 A380553 which equals x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.
%o A380553 (PARI) \\ As the Moebius transform of A118971 \\
%o A380553 {a(n) = sumdiv(n,d, moebius(n/d) * binomial(5*d-1,d-1)*4/(5*d-1) )}
%o A380553 for(n=1,30,print1(a(n),", "))
%o A380553 (PARI) \\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(4*n) ) \\
%o A380553 {a(n) = my(V=[0,1]); for(i=0,n, V = concat(V,0); A = Ser(V);
%o A380553 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 A380553 for(n=1,30,print1(a(n),", "))
%Y A380553 Cf. A346936, A034742, A380551, A380552, A118971, A002294, A008683.
%K A380553 nonn
%O A380553 1,2
%A A380553 _Paul D. Hanna_, Feb 16 2025