A260486 Expansion of phi(-x)^2 * phi(-x^6) / phi(-x^3) in powers of x where phi() is a Ramanujan theta function.

1, -4, 4, 2, -4, 0, 2, 0, -4, 0, 0, 8, -2, 0, 0, 0, -4, -8, 0, 8, 0, 0, 8, 0, -2, -4, 0, -2, 0, 0, 0, 0, -4, -4, 8, 0, 0, 0, 8, 0, 0, -8, 0, 8, -8, 0, 0, 0, -2, -4, 4, 4, 0, 0, -2, 0, 0, -4, 0, 8, 0, 0, 0, 0, -4, 0, 4, 8, -8, 0, 0, 0, 0, -8, 0, 2, -8, 0, 0, 0

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

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

Convolution of A010815 and A257657.

Example

G.f. = 1 - 4*x + 4*x^2 + 2*x^3 - 4*x^4 + 2*x^6 - 4*x^8 + 8*x^11 - 2*x^12 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^2 EllipticTheta[ 4, 0, x^6] / EllipticTheta[ 4, 0, x^3], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ x]^2 QPochhammer[ -x^3] / QPochhammer[ x^3], {x, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], -4 I^(n-1) Sum[ {1, I, -1/2, I, 1, -I/2}[[Mod[d, 6, 1]]] KroneckerSymbol[ -2, n/d], {d, Divisors[ n]}]];
a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -x]^2* EllipticTheta[3, 0, -x^6]/EllipticTheta[3, 0, -x^3], {x, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 17 2018 *)

Prog

(PARI) {a(n) = if( n<1, n==0, -4 * I^(n-1) * sumdiv(n, d, [-I/2, 1, I, -1/2, I, 1][d%6+1] * kronecker(-2, n/d)))};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n);  -4 * I^(n-1) * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, I, p==3, 1-e/2, p%8 > 4, !(e%2), e+1)))};
(PARI) {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A)^3 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^12 + A)), n))};

Crossrefs

Cf. A010815, A257657.

Sequence in context: A222295 A371706 A103714 * A193514 A112108 A373822

Adjacent sequences: A260483 A260484 A260485 * A260487 A260488 A260489

Keyword

sign

Author

Michael Somos, Jul 26 2015

Status

approved