A232343 - OEIS

%I #10 Sep 08 2022 08:46:06

%S 1,-1,2,0,3,-2,4,0,5,-5,8,0,7,-4,8,0,9,-8,10,0,14,-6,12,0,16,-14,14,0,

%T 15,-8,20,0,17,-14,18,0,19,-10,24,0,26,-21,22,0,23,-16,28,0,25,-20,32,

%U 0,32,-14,28,0,29,-28,30,0,38,-16,32,0,33,-31,40,0,40

%N Expansion of q^(-5/3) * c(q^2)^3 / (9 * c(q)) in powers of q where c() is a cubic AGM theta function.

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

%H G. C. Greubel, <a href="/A232343/b232343.txt">Table of n, a(n) for n = 0..2500</a>

%F Expansion of q^(-5/3) * eta(q) * eta(q^6)^9 / (eta(q^2) * eta(q^3))^3 in powers of q.

%F Euler transform of period 6 sequence [-1, 2, 2, 2, -1, -4, ...].

%F a(n) = 1/6 * b(3*n + 5) where b() is multiplicative with b(2^e) = 2 - 2^e, b(3^e) = 0^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

%F a(2*n) = A098098(n). a(4*n + 1) = - A033686(n). a(4*n + 3) = 0.

%e G.f. = 1 - x + 2*x^2 + 3*x^4 - 2*x^5 + 4*x^6 + 5*x^8 - 5*x^9 + 8*x^10 + ...

%e G.f. = q^5 - q^8 + 2*q^11 + 3*q^17 - 2*q^20 + 4*q^23 + 5*q^29 - 5*q^32 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^6]^9 / (QPochhammer[ x^2] QPochhammer[ x^3])^3, {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3] QPochhammer[ x^12])^3 / (QPochhammer[ -x] QPochhammer[ x^4]), {x, 0, n}];

%t a[ n_] := If[ n < 0, 0, Times @@ (Which[# == 2, 2 - 2^#2,# == 3, 1, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[3 n + 5]) / 6]; (* _Michael Somos_, Jul 09 2018 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^9 / (eta(x^2 + A) * eta(x^3 + A))^3, n))};

%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 5; A = factor(n); 1/6 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 2 - 2^e, p==3, 0, (p^(e+1) - 1) / (p - 1))))};

%o (Magma) Basis( ModularForms( Gamma0(18), 2), 210) [6]; /* _Michael Somos_, Jul 09 2018 */

%Y Cf. A033686, A098098.

%K sign

%O 0,3

%A _Michael Somos_, Nov 22 2013