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A258739
Expansion of (f(-x)^3 / f(-x^2))^6 - 64 * x * (f(-x^2)^3 / f(-x))^6 in powers of x where f() is a Ramanujan theta function.
1
1, -82, -243, -1194, 2242, 0, 3599, 2950, 0, -12242, -20950, 19926, -16807, 7294, 0, 18950, 97908, 0, -88806, 0, 59049, -183844, 51050, 0, -92142, -98002, 0, 246486, 118706, 290142, -161051, -38868, 0, 0, 75658, 0, -241900, 47614, -544806, -493658, 0, 0
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.
Denoted by g_6(q) in Cynk and Hulek on page 8 as a level 32 cusp form of weight 6.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/4) * ((eta(-q)^3 / eta(-q^2))^6 - 64 * (eta(-q^2) / eta(-q))^6) in powers of q.
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(5*e/2) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) if p == 1 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = -(32^3) (t/i)^6 f(t) where q = exp(2 Pi i t).
EXAMPLE
G.f. = 1 - 82*x - 243*x^2 - 1194*x^3 + 2242*x^4 + 3599*x^6 + 2950*x^7 + ...
G.f. = q - 82*q^5 - 243*q^9 - 1194*q^13 + 2242*q^17 + 3599*q^25 + 2950*q^29 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^3 / QPochhammer[ x^2])^6 - 64 x (QPochhammer[ x^2]^3 / QPochhammer[ x])^6, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^2 + A))^6 - 64 * x * (eta(x^2 + A)^3 / eta(x + A))^6, n))};
(PARI) {a(n) = my(A, p, e, x, y, a0, a1); if(n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 0, p%4==3, if(e%2, 0, (-p)^(5*e/2)), y = -sum(i=0, p-1, kronecker(i^3-i, p)); a0=2; a1=y; for(i=2, 5, x=y*a1 -p*a0; a0=a1; a1=x); y=a1; a0=1; a1=y; for(i=2, e, x=y*a1 -p^5*a0; a0=a1; a1=x); a1)))};
(Magma) A := Basis( CuspForms( Gamma0(32), 6), 165); A[1] - 82*A[5] - 243*A[9] - 1194*A[13] + 2242*A[16];
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jun 08 2015
STATUS
approved