A281786 Expansion of a(q) * b(q^2) + a(q^2) * b(q) in powers of q where a(), b() are cubic AGM functions.

2, 3, 3, -24, 3, 18, -24, 24, 3, -24, 18, 36, -24, 42, 24, -144, 3, 54, -24, 60, 18, -192, 36, 72, -24, 93, 42, -24, 24, 90, -144, 96, 3, -288, 54, 144, -24, 114, 60, -336, 18, 126, -192, 132, 36, -144, 72, 144, -24, 171, 93, -432, 42, 162, -24, 216, 24, -480

Offset: 0

Author

Michael Somos, Apr 09 2017

Keywords

• sign

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 2 + 3*q + 3*q^2 - 24*q^3 + 3*q^4 + 18*q^5 - 24*q^6 + 24*q^7 + 3*q^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A186099, A281785.

Programs

Mathematica

a[ n_] := If[ n < 1, 2 Boole[n == 0], 3 If[ Divisible[n, 3], -8, 1] DivisorSigma[ 1, n / (2^IntegerExponent[n, 2] 3^IntegerExponent[n, 3])]];
a[ n_] := If[ n < 1, 2 Boole[n == 0], 3 Times @@ (Which[ # < 3, 1, # == 3, -8, True, (#^(#2+1) - 1) / (# - 1)] & @@@ FactorInteger@n)];

PARI

{a(n) = if( n<1, 2*(n==0), if( n%3, 3, -24) * sigma(n / (2^valuation(n, 2) * 3^valuation(n, 3))))};

PARI

{a(n) = if( n<1, 2*(n==0), 3 * sumdiv(n, d, d*(d%2)) - if( n%3==0, 36 * sumdiv(n/3, d, d*(d%2))) + if( n%9==0, 81 * sumdiv(n/9, d, d*(d%2))))};

Formula

Expansion of 3 * b(q^3) * b(q^6) - b(q) * b(q^2) in powers of q where b() is a cubic AGM function.

a(n) = 3*b(n) if n>0 where b() is multiplicative with b(2^e) = 1, b(3^e) = -8 if e>0, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

a(n) = 3 * A281786(n) if n>0. a(2*n) = a(n). a(3*n) = -24 * A186099(n).