A161804 - OEIS

%I #6 Feb 11 2020 02:05:20

%S 1,3,3,12,30,27,66,141,111,255,513,378,903,1815,1356,2970,5727,4131,

%T 8571,15882,10881,23001,42417,29106,59763,108165,73500,145164,255831,

%U 167643,333693,585258,382053,751059,1302966,849339,1623009,2762349

%N G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 3*A038500(n) * q^n/n ).

%C A002129 forms the l.g.f. of log[ Sum_{n>=0} x^(n(n+1)/2) ], while 3*A038500 forms the l.g.f. of the log of the g.f. of A161809 and A038500(n) is the highest power of 3 dividing n.

%F Given trisections where A(q) = T_0(q^3) + q*T_1(q^3) + q^2*T_2(q^3):

%F T_0(q) = Sum_{n>=0} a(3n)*q^n,

%F T_1(q) = Sum_{n>=0} a(3n+1)*q^n,

%F T_2(q) = Sum_{n>=0} a(3n+2)*q^n,

%F then it appears that:

%F T_1(-q)/T_0(-q) = 3*q^(-1/3)*(eta(q^6)^4/(eta(q)*eta(q^3)*eta(q^4)*eta(q^12)))^2 (Cf. A132977);

%F T_2(-q)/T_0(-q) = 3*q^(-2/3)*(eta(q^2)*eta(q^6))^2*eta(q^3)*eta(q^12)/(eta(q)*eta(q^4))^3 (cf. A132978);

%F T_2(q)/T_1(q) = g.f. of A092848, the reciprocal of Hauptmodul for Gamma_0(18).

%e G.f.: A(q) = 1 + 3*q + 3*q^2 + 12*q^3 + 30*q^4 + 27*q^5 + 66*q^6 + ...

%e log(A(q)) = 3*q - 3*q^2 + 36*q^3 - 15*q^4 + 18*q^5 - 36*q^6 + 24*q^7 + ...

%e Sum_{n>=1} A002129(n)*q^n/n = log(1 + q + q^3 + q^6 + q^10 + q^15 + ...),

%e Sum_{n>=1} 3*A038500(n)*x^n/n = log of the g.f. of A161809.

%e TRISECTIONS:

%e T_0(q) = 1 + 12*q + 66*q^2 + 255*q^3 + 903*q^4 + 2970*q^5 + ... (A161805)

%e T_1(q) = 3 + 30*q + 141*q^2 + 513*q^3 + 1815*q^4 + 5727*q^5 + ... (A161806)

%e T_2(q) = 3 + 27*q + 111*q^2 + 378*q^3 + 1356*q^4 + 4131*q^5 + ... (A161807)

%e where T_1(-q)/T_0(-q)/3 equals (cf. A132977):

%e 1 + 2*q + 5*q^2 + 12*q^3 + 26*q^4 + 50*q^5 + 92*q^6 + 168*q^7 + ...

%e and T_2(-q)/T_0(-q)/3 equals (cf. A132978):

%e 1 + 3*q + 7*q^2 + 15*q^3 + 32*q^4 + 63*q^5 + 114*q^6 + 201*q^7 + ...

%e also, T_2(q)/T_1(q) equals (cf. A092848):

%e 1 - q + 2*q^3 - 2*q^4 - q^5 + 4*q^6 - 4*q^7 - q^8 + 8*q^9 - 8*q^10 + ...

%o (PARI) {a(n)=local(L=sum(m=1, n,3*3^valuation(m,3)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}

%Y Cf. trisections: A161805 (T_0), A161806 (T_1), A161807 (T_2).

%Y Cf. A132977 (T_1/T_0), A132978 (T_2/T_0), A092848 (T_2/T_1).

%Y Cf. A002129, A038500, A161809, A161800 (variant).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 20 2009