A168370 - OEIS

%I #2 Mar 30 2012 18:37:20

%S 1,3,36,2952,1670220,6781419711,204255279577440,47027922196061266047,

%T 84798672814179921118709052,1219732878003607687535196405346440,

%U 141916059665284234793191571472586402539060

%N a(n) = coefficient of x^n in the (3^n)-th power of 1 + Sum_{k>=0} x^(3^k) for n>=0.

%F G.f.: A(x) = Sum_{n>=0} log(F(3^n*x))^n/n! where F(x) = 1 + Sum_{n>=0} x^(3^n).

%e G.f.: A(x) = 1 + 3*x + 36*x^2 + 2952*x^3 + 1670220*x^4 +...

%e Let F(x) = 1 + x + x^3 + x^9 + x^27 + x^81 +...+ x^(3^n) +...

%e then A(x) = 1 + log(F(3x)) + log(F(9x))^2/2! + log(F(27x))^3/3! +...+ log(F(3^n*x))^n/n! +...

%e Also, coefficients in powers F(x)^(3^n) begin:

%e F^1:[(1),1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,...];

%e F^3:[1,(3),3,4,6,3,3,3,0,4,6,3,6,6,0,3,0,0,3,3,0,3,0,0,0,0,0,4,...];

%e F^9:[1,9,(36),93,198,378,624,918,1269,1606,1908,2277,2634,3060,...];

%e F^27:[1,27,351,(2952),18252,89505,366561,1300455,4101435,11713287,...];

%e F^81:[1,81,3240,85401,(1670220),25877556,331198416,3605580540,...];

%e F^243:[1,243,29403,2362284,141781266,(6781419711),269282151567,...];

%e F^729:[1,729,265356,64305333,11671816338,1692529329582,(204255279577440), ...]; ...

%e where the coefficients in parenthesis form the initial terms of this sequence.

%o (PARI) {a(n)=local(G=1+sum(m=0,ceil(log(n+3)/log(3)),x^(3^m))+x*O(x^n));polcoeff(G^(3^n),n)}

%o (PARI) {a(n)=local(G=1+sum(m=0,ceil(log(n+3)/log(3)),x^(3^m))+x*O(x^n));polcoeff(sum(m=0,n,log(subst(G,x,3^m*x))^m/m!),n)}

%Y Cf. A168369 (variant).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 24 2009