A214456 Expansion of b(q^2) * (b(q) + 2 * b(q^4)) / 3 in powers of q where b() is a cubic AGM theta function.

1, -1, -3, 5, -3, -6, 15, -8, -3, 23, -18, -12, 15, -14, -24, 30, -3, -18, 69, -20, -18, 40, -36, -24, 15, -31, -42, 77, -24, -30, 90, -32, -3, 60, -54, -48, 69, -38, -60, 70, -18, -42, 120, -44, -36, 138, -72, -48, 15, -57, -93, 90, -42, -54, 231, -72, -24, 100

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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (9 * phi(q^3)^4 - phi(q)^4) / 8 = phi(q) * (psi(-q) * chi(q^3))^3 = psi(-q)^4 * (chi(q) * chi(q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

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

Euler transform of period 12 sequence [ -1, -3, 2, -4, -1, -6, -1, -4, 2, -3, -1, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 36 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A214361.

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

G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^(6*k-3))^3 / (1 + x^(2*k-1)).

a(n) = (-1)^n * A118271(n). a(2*n) = a(4*n) = A131943(n). a(2*n + 1) = -A134077(n).

Example

1 - q - 3*q^2 + 5*q^3 - 3*q^4 - 6*q^5 + 15*q^6 - 8*q^7 - 3*q^8 + 23*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (9 EllipticTheta[ 3, 0, q^3]^4 - EllipticTheta[ 3, 0, q]^4) / 8, {q, 0, n}]
a[ n_] := SeriesCoefficient[ QPochhammer[q^2]^4 QPochhammer[-q^3, q^6]^3 /QPochhammer[-q, q^2], {q, 0, n}]
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] QPochhammer[ -q^3, q^6])^3 EllipticTheta[ 2, 0, (-q)^(1/2)]^4 / (16 (-q)^(1/2)), {q, 0, n}]

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A)^2 * eta(x^4 + A) * eta(x^6 + A)^6 / ( eta(x^3 + A) * eta(x^12 + A))^3, n))}
(PARI) {a(n) = if( n<1, n==0, -sigma(n) + if( n%3==0, 9 * sigma(n/3)) + if( n%4==0, 4 * sigma(n/4)) + if( n%12==0, -36 * sigma(n/12)))}
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 3, if( p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1))))))}

Crossrefs

Cf. A118271, A131943, A134077, A214361.

Sequence in context: A286108 A096438 A299418 * A118271 A260689 A328386

Adjacent sequences: A214453 A214454 A214455 * A214457 A214458 A214459

Keyword

sign

Author

Michael Somos, Jul 18 2012

Status

approved