A132003 Expansion of (phi(q^3) / phi(q)) * phi(-q^2) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

1, -2, 2, -2, 2, -4, 2, 0, 2, -2, 4, 0, 2, -4, 0, -4, 2, -4, 2, 0, 4, 0, 0, 0, 2, -6, 4, -2, 0, -4, 4, 0, 2, 0, 4, 0, 2, -4, 0, -4, 4, -4, 0, 0, 0, -4, 0, 0, 2, -2, 6, -4, 4, -4, 2, 0, 0, 0, 4, 0, 4, -4, 0, 0, 2, -8, 0, 0, 4, 0, 0, 0, 2, -4, 4, -6, 0, 0, 4, 0

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Equation (32.72).

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q)^2 * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^3) in powers of q.

a(n) = -2*b(n) where b() is multiplicative with b(2^e) = 2*0^e - 1, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).

Euler transform of period 12 sequence [-2, 1, 0, 0, -2, -4, -2, 0, 0, 1, -2, -2, ...].

G.f.: 1 - 2 * Sum_{k>0} Kronecker(-36, k) * x^k / (1 + x^k).

a(n) = - A132004(n) unless n=0.

a(2*n) = A122857(n). a(2*n + 1) = -2 * A125079(n). a(3*n) = a(n). a(3*n + 1) = -2 * A258277(n). a(3*n + 2) = 2 * A258278(n). - Michael Somos, Nov 01 2015

a(12*n + 7) = a(12*n + 11) = 0. a(4*n + 1) = -2 * A008441(n).

a(n) = (-1)^n * A122857(n). Expansion of (phi(-q)^2 + phi(-q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 05 2023

Example

G.f. = 1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 4*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + 4*x^10 + ...

Mathematica

a[ n_] := If[ n < 1, Boole[n == 0], -2 DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -2 Times @@ (Which[ # < 5, -(-1)^#, Mod[#, 4] == 3, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^6] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q]^2 + EllipticTheta[ 4, 0, q^3]^2) / 2, {q, 0, n}]; (* Michael Somos, Mar 05 2023 *)

Prog

(PARI) {a(n) = if( n<1, n==0, -2 * sumdiv(n, d, (-1)^(n+d) * kronecker(-36, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^3), n))};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, p==2, -1, p%4==1, e+1, 1-e%2)))};

Crossrefs

Cf. A008441, A122857, A125079, A132004.

Sequence in context: A370690 A062816 A306653 * A122857 A109810 A365498

Adjacent sequences: A132000 A132001 A132002 * A132004 A132005 A132006

Keyword

sign

Author

Michael Somos, Aug 06 2007

Status

approved