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A255258
Expansion of q^2 * phi(q) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.
2
1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0
OFFSET
2,2
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^32)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^16)) in powers of q.
Euler transform of period 32 sequence [ 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A224609.
(-1)^n * a(n) = A227395(n).
a(4*n) = a(4*n + 1) = a(8*n + 7) = 0. a(4*n + 2) = A113411(n). a(8*n + 3) = 2 * A033761(n).
EXAMPLE
G.f. = q^2 + 2*q^3 + 2*q^6 + 2*q^11 + 3*q^18 + 2*q^19 + 2*q^22 + 4*q^27 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^8] / 2, {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^32 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)), n))};
(Magma) A := Basis( ModularForms( Gamma1(32), 1), 89); A[3] + 2*A[4] + 2*A[7] + 2*A[12];
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Feb 19 2015
STATUS
approved