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A256276
Expansion of q * phi(q) * chi(q^3) * psi(-q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
4
1, 2, 0, 1, 4, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 4, 0, 0, 0, 0, 3, 4, 0, 0, 4, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 0, 0, 0, 0, 1, 6, 0, 2, 4, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 8, 0, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 4, 0, 0
OFFSET
1,2
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^6)^2 * eta(q^9) * eta(q^36) / (eta(q)^2 * eta(q^4)^2 * eta(q^3) * eta(q^12) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 2, -3, 3, -1, 2, -4, 2, -1, 2, -3, 2, -1, 2, -3, 3, -1, 2, -4, 2, -1, 3, -3, 2, -1, 2, -3, 2, -1, 2, -4, 2, -1, 3, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A256269.
a(3*n) = a(4*n + 3) = 0. a(3*n + 1) = A122865(n). a(3*n + 2) = 2 * A122856(n). a(4*n + 1) = a(n). a(4*n) = a(n). a(6*n + 2) = 2 * A122865(n). a(6*n + 4) = A122856(n).
EXAMPLE
G.f. = q + 2*q^2 + q^4 + 4*q^5 + 2*q^8 + 2*q^10 + 2*q^13 + q^16 + 4*q^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(9/2)] / (2^(1/2) q^(1/8)) QPochhammer[ -q^3, q^6] EllipticTheta[ 3, 0, q], {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^3 + A) * eta(x^12 + A) * eta(x^18 + A)), n))};
(Magma) A := Basis( ModularForms( Gamma1(36), 1), 89); A[2] + 2*A[3]
+ A[5] + 4*A[6] + 2*A[9] + 2*A[11] + 2*A[14] + A[17] + 4*A[18];
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jun 02 2015
STATUS
approved