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A270640 Expansion of psi(x) * psi(x^6) / psi(-x^3) in powers of x where psi() is a Ramanujan theta function. 1
1, 1, 0, 2, 1, 0, 4, 2, 0, 6, 4, 0, 9, 5, 0, 14, 8, 0, 20, 12, 0, 30, 16, 0, 41, 22, 0, 56, 32, 0, 76, 42, 0, 102, 56, 0, 136, 75, 0, 178, 97, 0, 232, 126, 0, 300, 164, 0, 384, 208, 0, 490, 264, 0, 620, 336, 0, 780, 420, 0, 977, 526, 0, 1218, 656, 0, 1512, 810 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
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 psi(x) / (chi(-x^3) * chi(-x^6)) in powers of x where psi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q^2)^2 * eta(q^12) / (eta(q) * eta(q^3)) in powers of q.
Euler transform of period 12 sequence [ 1, -1, 2, -1, 1, 0, 1, -1, 2, -1, 1, -1, ...].
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 + x^(6*k)).
EXAMPLE
G.f. = 1 + x + 2*x^3 + x^4 + 4*x^6 + 2*x^7 + 6*x^9 + 4*x^10 + 9*x^12 + ...
G.f. = q + q^3 + 2*q^7 + q^9 + 4*q^13 + 2*q^15 + 6*q^19 + 4*q^21 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 2^-1 x^(-1/8) EllipticTheta[ 2, 0, x^(1/2)] / (QPochhammer[ x^3, x^6] QPochhammer[ x^6, x^12]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^3] / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A) / (eta(x + A) * eta(x^3 + A)), n))};
CROSSREFS
Sequence in context: A136329 A122073 A106236 * A139136 A122792 A348218
KEYWORD
nonn
AUTHOR
Michael Somos, Mar 20 2016
STATUS
approved

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Last modified April 12 07:53 EDT 2024. Contains 371626 sequences. (Running on oeis4.)