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A229615 Expansion of q^2 * psi(q^3)^6 / psi(q)^2 in powers of q where psi() is a Ramanujan theta function. 4
1, -2, 3, 0, -1, 0, 7, -8, 6, 0, 1, 0, 8, -12, 15, 0, -7, 0, 18, -16, 12, 0, 5, 0, 14, -26, 24, 0, -6, 0, 31, -24, 18, 0, -5, 0, 20, -28, 42, 0, -8, 0, 36, -48, 24, 0, 13, 0, 31, -36, 42, 0, -25, 0, 56, -40, 30, 0, 6, 0, 32, -64, 63, 0, -12, 0, 54, -48, 48, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

LINKS

G. C. Greubel, Table of n, a(n) for n = 2..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of (a(q) - a(q^2))^2 / 36 = c(q^2)^4 / (9 * c(q)^2) in powers of q where a(), c() are cubic AGM theta functions.

Expansion of ((eta(q) * eta(q^6)^6) / (eta(q^2)^2 * eta(q^3)^3))^2 in powers of q.

Euler transform of period 6 sequence [ -2, 2, 4, 2, -2, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/12) (t / i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229616.

G.f.: sum_{k>0} x^(6*k-4) / (1 - x^(6*k-4))^2 - 2 * x^(6*k-3) / (1 - x^(6*k-3))^2 + x^(6*k-2) / (1 - x^(6*k-2))^2.

G.f.: sum_{k>0} (3*k-2) * x^(6*k-4) / (1 - x^(6*k-4)) - (4*k-2) * x^(6*k-3) / (1 - x^(6*k-3)) + (3*k-1) * x^(6*k-2) / (1 - x^(6*k-2)).

a(6*n + 1) = a(6*n + 5) = 0. a(6*n + 2) = A144614(n). a(6*n + 3) = -2 * A008438(n). a(6*n + 4) = 3 * A033686(n).

Convolution square of A093829.

EXAMPLE

G.f. = q^2 - 2*q^3 + 3*q^4 - q^6 + 7*q^8 - 8*q^9 + 6*q^10 + q^12 + ...

MATHEMATICA

a[ n_] := If[n < 1, 0, Sum[ {0, 1, -2, 1, 0, 0}[[ Mod[d, 6, 1]]] n/d, {d, Divisors[n]}]];

a[ n_] := If[n < 1, 0, Sum[ {0, 1/2, -2/3, 1/2, 0, 0}[[ Mod[d, 6, 1]]] d, {d, Divisors[n]}]];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(3/2)]^6 / EllipticTheta[ 2, 0, q^(1/2)]^2 / 16, {q, 0, n}];

PROG

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n/d * [0, 0, 1, -2, 1, 0][d%6 + 1]))};

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d * [0, 0, 1/2, -2/3, 1/2, 0][d%6 + 1]))};

(PARI) {a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^6 / (eta(x^2 + A)^2 * eta(x^3 + A)^3))^2, n))};

(Sage) ModularForms( Gamma0(6), 2, prec=70).2;

CROSSREFS

Cf. A008438, A033686, A093829, A144614, A229616.

Sequence in context: A298159 A123735 A155839 * A135814 A038570 A306559

Adjacent sequences:  A229612 A229613 A229614 * A229616 A229617 A229618

KEYWORD

sign

AUTHOR

Michael Somos, Sep 26 2013

STATUS

approved

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Last modified March 26 05:43 EDT 2019. Contains 321481 sequences. (Running on oeis4.)