cp's OEIS Frontend

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

%I A380551 #16 Feb 17 2025 17:38:07
%S A380551 1,1,6,28,142,720,3875,21288,120168,690546,4032014,23840724,142498691,
%T A380551 859512043,5225263875,31983651216,196947587822,1219199232294,
%U A380551 7583142491924,47365473951152,296983176365613,1868545308601424,11793499763070479,74650344221104632,473770694965305205,3014124873709172435
%N A380551 G.f. A(x) satisfies x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ).
%C A380551 Moebius transform of A006013.
%H A380551 Paul D. Hanna, <a href="/A380551/b380551.txt">Table of n, a(n) for n = 1..500</a>
%F A380551 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A380551 (1) x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ).
%F A380551 (2) x = Sum_{n>=1} a(n) * x^n*(1-x)^(2*n) / (1 - x^n*(1-x)^(2*n)).
%F A380551 (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.
%F A380551 (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.
%e A380551 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 + ...
%e A380551 where x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ).
%e A380551 RELATED SERIES.
%e A380551 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) + ...
%e A380551 which equals x*F(x)^2 where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
%o A380551 (PARI) \\ As the Moebius transform of A006013 \\
%o A380551 {a(n) = sumdiv(n,d, moebius(n/d) * binomial(3*d-1,d-1)*2/(3*d-1) )}
%o A380551 for(n=1,30,print1(a(n),", "))
%o A380551 (PARI) \\ By definition x = Sum_{n>=1} A( x^n*(1-x)^(2*n) ) \\
%o A380551 {a(n) = my(V=[0,1]); for(i=0,n, V = concat(V,0); A = Ser(V);
%o A380551 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]}
%o A380551 for(n=1,30,print1(a(n),", "))
%Y A380551 Cf. A346925, A034742, A380552, A380553, A006013, A001764, A008683.
%K A380551 nonn
%O A380551 1,3
%A A380551 _Paul D. Hanna_, Feb 16 2025