A246815 Expansion of phi(-x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, -2, -2, 4, 3, -2, -6, 4, 4, -6, -4, 4, 7, -8, -2, 8, 8, -4, -10, 4, 4, -10, -10, 8, 9, -4, -6, 12, 8, -6, -10, 12, 4, -14, -8, 4, 16, -10, -8, 8, 9, -10, -12, 12, 8, -12, -12, 4, 20, -10, -6, 20, 8, -6, -10, 12, 8, -20, -18, 8, 11, -12, -12, 16, 8, -6, -20

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

Euler transform of period 8 sequence [ -2, -3, -2, -1, -2, -3, -2, -3, ...].

a(n) = (-1)^floor((n+1) / 2) * A045828(n).

a(4*n) = A213622(n). a(4*n + 3) = 4 * A033763(n).

Example

G.f. = 1 - 2*x - 2*x^2 + 4*x^3 + 3*x^4 - 2*x^5 - 6*x^6 + 4*x^7 + 4*x^8 + ...
G.f. = q - 2*q^3 - 2*q^5 + 4*q^7 + 3*q^9 - 2*q^11 - 6*q^13 + 4*q^15 + ...

Mathematica

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

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A) * eta(x^8 + A)^2 / eta(x^4 + A)^2, n))};

Crossrefs

Cf. A033763, A045828, A213622.

Sequence in context: A316774 A333920 A356164 * A246836 A246953 A045828

Adjacent sequences: A246812 A246813 A246814 * A246816 A246817 A246818

Keyword

sign

Author

Michael Somos, Sep 03 2014

Status

approved