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A298931
Expansion of psi(x^4) * c(x^3) / (3*x) where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.
3
1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 2, 0, 3, 0, 0, 2, 2, 0, 4, 1, 0, 0, 2, 0, 4, 0, 0, 4, 1, 0, 6, 2, 0, 0, 2, 0, 5, 0, 0, 3, 3, 0, 6, 1, 0, 0, 4, 0, 6, 0, 0, 4, 5, 0, 4, 3, 0, 0, 2, 0, 8, 0, 0, 4, 3, 0, 6, 3, 0, 0, 4, 0, 9, 0, 0, 6, 4, 0, 6, 2, 0, 0, 4, 0, 6, 0, 0
OFFSET
0,7
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-3/2) * eta(q^8)^2 * eta(q^9)^3 / (eta(q^3) * eta(q^4)) in powers of q.
Euler transform of a period 72 sequence.
A005872(2*n + 3) = 6*a(n). a(3*n) = A298932(n). a(3*n + 1) = A263452(n-1). a(3*n + 2) = a(4*n + 1) = 0.
EXAMPLE
G.f. = q^3 + q^9 + q^11 + 2*q^15 + q^17 + 2*q^23 + 3*q^27 + 2*q^33 + ...
G.f. = 1 + x^3 + x^4 + 2*x^6 + x^7 + 2*x^10 + 3*x^12 + 2*x^15 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^8]^2 QPochhammer[ x^9]^3 / (QPochhammer[ x^3] QPochhammer[ x^4]), {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 * eta(x^9 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 29 2018
STATUS
approved