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A213592
Expansion of q^(-1/3) * phi(q^2) * c(q) / 3 in powers of q where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.
4
1, 1, 4, 2, 6, 1, 6, 2, 7, 4, 8, 4, 10, 2, 10, 0, 9, 6, 12, 6, 10, 1, 14, 4, 16, 6, 8, 8, 12, 2, 12, 0, 20, 7, 20, 6, 10, 4, 20, 6, 11, 8, 16, 8, 20, 4, 14, 0, 20, 10, 12, 8, 26, 2, 22, 6, 15, 10, 20, 12, 18, 0, 28, 0, 20, 9, 20, 14, 16, 6, 10, 6, 24, 12, 32
OFFSET
0,3
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 q^(-1/3) * eta(q^3)^3 * eta(q^4)^5 / (eta(q) * eta(q^2)^2 * eta(q^8)^2) in powers of q.
Euler transform of period 24 sequence [ 1, 3, -2, -2, 1, 0, 1, 0, -2, 3, 1, -5, 1, 3, -2, 0, 1, 0, 1, -2, -2, 3, 1, -3, ...].
a(16*n + 15) = 0. a(4*n + 1) = a(n).
EXAMPLE
1 + x + 4*x^2 + 2*x^3 + 6*x^4 + x^5 + 6*x^6 + 2*x^7 + 7*x^8 + 4*x^9 + ...
q + q^4 + 4*q^7 + 2*q^10 + 6*q^13 + q^16 + 6*q^19 + 2*q^22 + 7*q^25 + ...
MATHEMATICA
QP := QPochhammer; a[n_]:= SeriesCoefficient[(QP[q^3]^3*QP[q^4]^5)/( QP[q]*QP[q^2]^2*QP[q^8]^2), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 07 2018 *)
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A)^5 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^8 + A)^2), n))}
CROSSREFS
Sequence in context: A131749 A016515 A011306 * A019719 A205111 A340177
KEYWORD
nonn
AUTHOR
Michael Somos, Jun 15 2012
STATUS
approved