A161804 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 3*A038500(n) * q^n/n ).

1, 3, 3, 12, 30, 27, 66, 141, 111, 255, 513, 378, 903, 1815, 1356, 2970, 5727, 4131, 8571, 15882, 10881, 23001, 42417, 29106, 59763, 108165, 73500, 145164, 255831, 167643, 333693, 585258, 382053, 751059, 1302966, 849339, 1623009, 2762349

Offset: 0

Author

Paul D. Hanna, Jul 20 2009

Keywords

• nonn

Comments

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.

Examples

			G.f.: A(q) = 1 + 3*q + 3*q^2 + 12*q^3 + 30*q^4 + 27*q^5 + 66*q^6 + ...
log(A(q)) = 3*q - 3*q^2 + 36*q^3 - 15*q^4 + 18*q^5 - 36*q^6 + 24*q^7 + ...
Sum_{n>=1} A002129(n)*q^n/n = log(1 + q + q^3 + q^6 + q^10 + q^15 + ...),
Sum_{n>=1} 3*A038500(n)*x^n/n = log of the g.f. of A161809.
TRISECTIONS:
T_0(q) = 1 + 12*q + 66*q^2 + 255*q^3 + 903*q^4 + 2970*q^5 + ... (A161805)
T_1(q) = 3 + 30*q + 141*q^2 + 513*q^3 + 1815*q^4 + 5727*q^5 + ... (A161806)
T_2(q) = 3 + 27*q + 111*q^2 + 378*q^3 + 1356*q^4 + 4131*q^5 + ... (A161807)
where T_1(-q)/T_0(-q)/3 equals (cf. A132977):
1 + 2*q + 5*q^2 + 12*q^3 + 26*q^4 + 50*q^5 + 92*q^6 + 168*q^7 + ...
and T_2(-q)/T_0(-q)/3 equals (cf. A132978):
1 + 3*q + 7*q^2 + 15*q^3 + 32*q^4 + 63*q^5 + 114*q^6 + 201*q^7 + ...
also, T_2(q)/T_1(q) equals (cf. A092848):
1 - q + 2*q^3 - 2*q^4 - q^5 + 4*q^6 - 4*q^7 - q^8 + 8*q^9 - 8*q^10 + ...
		

Crossrefs

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

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

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

Programs

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)}

Formula

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

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

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

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

then it appears that:

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);

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);

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