A216713 - OEIS

%I #8 Sep 14 2012 17:15:13

%S 1,1,3,12,27,105,420,1242,5295,22395,72738,323268,1410684,4806675,

%T 21881721,97371786,341608239,1579726122,7123796790,25489388367,

%U 119184247992,542664427242,1969440159591,9284827569117,42584603672868,156213604844883,741154831030785

%N G.f.: A(x) = 1 + x*A(x)^2 / ( A(w*x)*A(w^2*x) ), where w = exp(2*Pi*I/3).

%H Paul D. Hanna, <a href="/A216713/b216713.txt">Table of n, a(n) for n = 0..200</a>

%e G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 27*x^4 + 105*x^5 + 420*x^6 +...

%e Related expansions:

%e A(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 87*x^4 + 336*x^5 + 1356*x^6 +...

%e A(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 189*x^4 + 756*x^5 + 3132*x^6 +...

%e Let w = exp(2*Pi*I/3), then A(x) = 1 + x*A(x)^3/(A(x)*A(w*x)*A(w^2*x)) where

%e A(x)*A(w*x)*A(w^2*x) = 1 + 28*x^3 + 1134*x^6 + 61857*x^9 + 3929121*x^12 + 272388420*x^15 + 19981576476*x^18 + 1524888581787*x^21 +...

%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=1,n+1,A=1+x*A^3*exp(-3*sum(m=1,n\3,x^(3*m)*polcoeff(log(A),3*m))+x*O(x^n)));polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A143339, A212527, A216712, A216681.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 14 2012