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A227229
Expansion of (psi(q)^3 / psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.
2
1, 6, 15, 24, 33, 36, 33, 48, 69, 78, 90, 72, 51, 84, 120, 144, 141, 108, 87, 120, 198, 192, 180, 144, 87, 186, 210, 240, 264, 180, 198, 192, 285, 288, 270, 288, 105, 228, 300, 336, 414, 252, 264, 264, 396, 468, 360, 288, 159, 342, 465, 432, 462, 324, 249
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Number 8 and 32 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018
LINKS
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) + a(q^2))^2 / 4 in powers of q where a() is a cubic AGM theta function.
Expansion of (b(q^2)^2 / b(q))^2 in powers of q where b() is a cubic AGM theta function.
Expansion of (eta(q^3) * eta(q^2)^6 / (eta(q)^3 * eta(q^6)^2))^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227226.
Convolution square of A107760.
Euler transform of period 6 sequence [6, -6, 4, -6, 6, -4, ...].
EXAMPLE
G.f. = 1 + 6*q + 15*q^2 + 24*q^3 + 33*q^4 + 36*q^5 + 33*q^6 + 48*q^7 + 69*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^3] QPochhammer[ q^2]^6 / (QPochhammer[q]^3 QPochhammer[q^6]^2))^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^6 / EllipticTheta[ 2, 0, q^3]^2/16, {q, 0, 2 n}];
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 3/2, 2, 3/2, 2, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];
a[ n_] := If[ n < 1, Boole[ n == 0], 3 Sum[ {2, 1, 2, 1, 2, -8}[[ 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)^6 * eta(x^3 + A) / (eta(x + A)^3 * eta(x^6 + A)^2))^2, n))};
(Sage) A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[0] + 6*A[1] + 15*A[2];
(Magma) A := Basis( ModularForms( Gamma0(6), 2), 50); A[1] + 6*A[2] + 15*A[3];
CROSSREFS
Sequence in context: A238905 A187918 A190747 * A274319 A043477 A055040
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 19 2013
STATUS
approved