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

%I A281786 #8 Jul 29 2018 20:45:07
%S A281786 2,3,3,-24,3,18,-24,24,3,-24,18,36,-24,42,24,-144,3,54,-24,60,18,-192,
%T A281786 36,72,-24,93,42,-24,24,90,-144,96,3,-288,54,144,-24,114,60,-336,18,
%U A281786 126,-192,132,36,-144,72,144,-24,171,93,-432,42,162,-24,216,24,-480
%N A281786 Expansion of a(q) * b(q^2) + a(q^2) * b(q) in powers of q where a(), b() are cubic AGM functions.
%C A281786 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A281786 G. C. Greubel, <a href="/A281786/b281786.txt">Table of n, a(n) for n = 0..5000</a>
%F A281786 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.
%F A281786 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.
%F A281786 a(n) = 3 * A281786(n) if n>0. a(2*n) = a(n). a(3*n) = -24 * A186099(n).
%e A281786 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 + ...
%t A281786 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])]];
%t A281786 a[ n_] := If[ n < 1, 2 Boole[n == 0], 3 Times @@ (Which[ # < 3, 1, # == 3, -8, True, (#^(#2+1) - 1) / (# - 1)] & @@@ FactorInteger@n)];
%o A281786 (PARI) {a(n) = if( n<1, 2*(n==0), if( n%3, 3, -24) * sigma(n / (2^valuation(n, 2) * 3^valuation(n, 3))))};
%o A281786 (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))))};
%Y A281786 Cf. A186099, A281785.
%K A281786 sign
%O A281786 0,1
%A A281786 _Michael Somos_, Apr 09 2017