A161808 - OEIS

%I #11 Feb 18 2019 23:47:15

%S 1,3,3,3,9,12,12,27,36,57,141,165,135,321,450,399,780,1068,1308,2913,

%T 3537,2736,5940,8430,7173,13251,18267,17661,35007,45051,31866,58506,

%U 85890,65694,102000,145293,101547,140574,203781,114765,93051,161754

%N G.f.: A(q) = exp( Sum_{n>=1} A162552(n) * 3*A038500(n) * q^n/n ).

%C A162552 forms the l.g.f. of log[ Sum_{n>=0} x^(n^2) ], and

%C A038500(n) is the highest power of 3 dividing n.

%C The first negative term is a(43) = -162729.

%H Paul D. Hanna, <a href="/A161808/b161808.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..100 from Georg Fischer)

%e G.f.: A(q) = 1 + 3*q + 3*q^2 + 3*q^3 + 9*q^4 + 12*q^5 + 12*q^6 +...

%e log(A(q)) = 3*q - 3*q^2/2 + 9*q^3/3 + 9*q^4/4 - 12*q^5/5 + 45*q^6/6 - 18*q^7/7 +...

%e Compare to: q - q^2/2 + q^3/3 + 3*q^4/4 - 4*q^5/5 + 5*q^6/6 - 6*q^7/7 +...

%e which equals log( Sum_{n>=0} q^(n^2) ) as described by A162552.

%o (PARI) {a(n)=local(Q=sum(m=0,n,x^(m^2))+x*O(x^n),A); A=exp(sum(k=1,n,polcoeff(log(Q),k)*3*3^valuation(k,3)*x^k)+x*O(x^n));polcoeff(A,n)}

%Y Cf. A161804 (variant).

%K sign

%O 0,2

%A _Paul D. Hanna_, Jul 21 2009