OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.
FORMULA
Expansion of (a(q) + 2*a(q^2))^2 / 9 in powers of q where a(q) is a cubic AGM theta function.
Expansion of c(q)^4 / (3 * c(q^2))^2 in powers of q where c(q) is a cubic AGM theta function.
Expansion of (eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3))^2 in powers of q.
Euler transform of period 6 sequence [4, 2, -8, 2, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227229.
Convolution square of A123330.
EXAMPLE
G.f. = 1 + 4*q + 12*q^2 + 20*q^3 + 28*q^4 + 24*q^5 + 28*q^6 + 32*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^3]^6 / (QPochhammer[ q]^2 QPochhammer[ q^6]^3))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^6 / EllipticTheta[ 4, 0, q]^2, {q, 0, n}];
a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ {1, 1, 4/3, 1, 1, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := If[ n < 1, Boole[ n == 0], 4 Sum[ {1, 1, 2, 1, 1, -6}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^6 / (eta(x + A)^2 * eta(x^6 + A)^3))^2, n))};
(Sage) A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 4*A[1] + 12*A[2];
(Magma) A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 4*A[2] + 12*A[3];
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 19 2013
STATUS
approved