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A260109 Expansion of f(x^3) * f(-x^3)^2 * psi(x)^2 / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions. 3
1, 3, 4, 6, 9, 12, 14, 12, 16, 18, 18, 24, 21, 27, 28, 30, 36, 24, 38, 42, 40, 42, 36, 48, 43, 48, 52, 48, 54, 60, 62, 54, 56, 66, 72, 72, 74, 63, 72, 78, 81, 84, 64, 84, 88, 84, 90, 72, 98, 108, 100, 102, 72, 108, 110, 114, 112, 96, 126, 96, 133, 120, 104 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 = 0..2500

FORMULA

Expansion of psi(-x^3) * phi(-x^6)^2 * psi(x)^2 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of phi(x) * f(x, x^5) * f(x^2, x^4)^2 in powers of x. - Michael Somos, Jul 18 2015

Expansion of q^(-1/2) * eta(q^2)^5 * eta(q^3) * eta(q^6)^3 / (eta(q)^3 * eta(q^4) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [ 3, -2, 2, -1, 3, -6, 3, -1, 2, -2, 3, -4, ...].

a(n) = A124815(2*n + 1). a(3*n + 1) = 3 * a(n).

EXAMPLE

G.f. = 1 + 3*x + 4*x^2 + 6*x^3 + 9*x^4 + 12*x^5 + 14*x^6 + 12*x^7 + ...

G.f. = q + 3*q^3 + 4*q^5 + 6*q^7 + 9*q^9 + 12*q^11 + 14*q^13 + 12*q^15 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1/4 x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 4, 0, x^6]^2 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}];

a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/8) QPochhammer[ -x^3] QPochhammer[ x^3]^2 EllipticTheta[ 2, 0, x^(1/2)]^2 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^6 + A)^3 / (eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A)), n))};

CROSSREFS

Cf. A124815.

Sequence in context: A022297 A010430 A191287 * A283777 A202171 A182531

Adjacent sequences:  A260106 A260107 A260108 * A260110 A260111 A260112

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 16 2015

STATUS

approved

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Last modified January 20 10:30 EST 2022. Contains 350471 sequences. (Running on oeis4.)