A260313 - OEIS

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A260313

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

1, 3, 1, -2, 3, 4, -3, -3, 2, 7, 0, -9, 4, 9, -5, -11, 6, 18, -7, -18, 9, 20, -12, -27, 14, 36, -11, -42, 18, 46, -24, -54, 23, 69, -27, -79, 37, 90, -44, -104, 48, 126, -52, -147, 65, 162, -78, -189, 85, 225, -91, -254, 114, 286, -136, -327, 142, 381, -159

(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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(-x) * chi(x)^4 = phi(-x^2) * chi(x)^3 = psi(x)^3 / psi(x^2)^2 = phi(-x^2)^4 / f(-x)^3 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.

Expansion of q^(1/8) * eta(q^2)^8 / (eta(q)^3 * eta(q^4)^4) in powers of q.

Euler transform of period 4 sequence [ 3, -5, 3, -1, ...].

a(n) = A153172(n) + 2*A153174(n).

EXAMPLE

G.f. = 1 + 3*x + x^2 - 2*x^3 + 3*x^4 + 4*x^5 - 3*x^6 - 3*x^7 + 2*x^8 + ...

G.f. = 1/q + 3*q^7 + q^15 - 2*q^23 + 3*q^31 + 4*q^39 - 3*q^47 - 3*q^55 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 2 x^(1/8) / EllipticTheta[ 2, 0, 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)^8 / (eta(x + A)^3 * eta(x^4 + A)^4), n))};

CROSSREFS

Cf. A153172, A153174.