A246962 Expansion of psi(-x^3) * phi(-x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 0, -2, -1, 0, 2, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, -2, -1, 0, -2, 2, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 0, -2, -2, 0, 2, 0, 0, -2, 0, 0, 0, -1, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 0, -2, 0, 0, -2, 2, 0, 0, -2, 0, -2, 0, 0, -2

Offset

0,3

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 q^(-3/8) * eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^12)) in powers of q.

Euler transform of period 12 sequence [0, -2, -1, -1, 0, -2, 0, -1, -1, -2, 0, -2, ...].

a(3*n) = A226860(n). a(3*n + 1) = 0.

a(3*n + 2) = -2 * A257469(n). - Michael Somos, Apr 25 2015

Example

G.f. = 1 - 2*x^2 - x^3 + 2*x^5 + 2*x^8 - x^9 - 2*x^17 - x^18 - 2*x^20 + ...
G.f. = q^3 - 2*q^19 - q^27 + 2*q^43 + 2*q^67 - q^75 - 2*q^139 - q^147 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)] / (Sqrt[2] q^(3/8)), {q, 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^3 + A) * eta(x^12 + A) / (eta(x^4 + A) * eta(x^6 + A)), n))};
(PARI) {a(n) = my(A, p, e, i); if( n<0, 0, n = 8*n + 3; A = factor(n); -I * prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==2, 0, if( p==3, I^e, if( p%24 == 1 || p%24==19, for(j=1, sqrtint(p\18), if( issquare( p - 18*j^2, &i), break)); (e+1) * (if(p%24==1, 1, -I) * kronecker( 12, i))^e, if( e%2, 0, if(p%24>12, 1, -1)^(e/2)))) ))))};

Crossrefs

Cf. A226860, A257469.

Sequence in context: A193275 A182033 A112214 * A112608 A058677 A262780

Adjacent sequences: A246959 A246960 A246961 * A246963 A246964 A246965

Keyword

sign

Author

Michael Somos, Sep 08 2014

Status

approved