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A204531
Expansion of phi(q) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.
5
1, 2, 0, 0, 0, -4, 0, 0, -4, 2, 0, 0, 0, -4, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, -4, 0, 0, 4, 0, 0, 0, 0, -4, 0, 0, -8, 4, 0, 0, 0, -4, 0, 0, 0, 2, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 0, -8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^2)^5 / (eta(q)^2 * eta(q^8)) in powers of q.
Euler transform of period 8 sequence [ 2, -3, 2, -3, 2, -3, 2, -2, ...].
G.f.: Product_{k>0} (1 - x^(2*k))^5 / ((1 - x^k)^2 * (1 - x^(8*k))).
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0. a(8*n) = A104794(n). a(4*n + 1) = 2 * A134343(n).
a(n) = (-1)^n * A246950(n). a(8*n + 1) = 2 * A113407(n). a(8*n + 5) = -4 * A053692(n). - Michael Somos, Jun 10 2015
EXAMPLE
G.f. = 1 + 2*q - 4*q^5 - 4*q^8 + 2*q^9 - 4*q^13 + 4*q^16 + 4*q^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^4], {q, 0, n}]; (* Michael Somos, Jun 10 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^8 + A)), n))};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * (-1)^(n%8==5) * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 2 * (e>2) * (-1)^(e<4), p%4==1, e+1, !(e%2))))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jan 15 2012
STATUS
approved