A226535 Expansion of b(-q) in powers of q where b() is a cubic AGM theta function.

1, 3, 0, -6, -3, 0, 0, 6, 0, -6, 0, 0, 6, 6, 0, 0, -3, 0, 0, 6, 0, -12, 0, 0, 0, 3, 0, -6, -6, 0, 0, 6, 0, 0, 0, 0, 6, 6, 0, -12, 0, 0, 0, 6, 0, 0, 0, 0, 6, 9, 0, 0, -6, 0, 0, 0, 0, -12, 0, 0, 0, 6, 0, -12, -3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, -6, -6, 0, 0, 6, 0

Offset

0,2

Comments

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

Zagier (2009) denotes the g.f. as f(z) in Case B which is associated with F(t) the g.f. of A006077.

References

D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

Links

Seiichi Manyama, 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

D. Zagier, Integral solutions of Apery-like recurrence equations.

Formula

Expansion of f(q)^3 / f(q^3) in powers of q where f() is a Ramanujan theta function.

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

Expansion of eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6))^3 in powers of q.

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

Moebius transform is period 36 sequence [ 3, -3, -9, -3, -3, 9, 3, 3, 0, 3, -3, 9, 3, -3, 9, -3, -3, 0, 3, 3, -9, 3, -3, -9, 3, -3, 0, -3, -3, -9, 3, 3, 9, 3, -3, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227696.

G.f.: f(q) = F(t(q)) where F() is the g.f. of A006077 and t() is the g.f. of A227454.

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

a(3*n + 2) = a(4*n + 2) = 0.

a(n) = (-1)^n * A005928(n) = (-1)^(((n+1) mod 6 ) > 3) * A113062(n). A113062(n) = |a(n)|.

a(3*n) = A180318(n). a(2*n + 1) = 3 * A123530(n). a(4*n) = A005928(n).

Example

G.f. = 1 + 3*q - 6*q^3 - 3*q^4 + 6*q^7 - 6*q^9 + 6*q^12 + 6*q^13 - 3*q^16 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^3 / QPochhammer[ -q^3], {q, 0, n}]

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^3, n))}

Crossrefs

Cf. A005928, A006077, A113062, A180318, A227454, A227696.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

Sequence in context: A334874 A343903 A339416 * A005928 A113062 A259659

Adjacent sequences: A226532 A226533 A226534 * A226536 A226537 A226538

Keyword

sign

Author

Michael Somos, Sep 22 2013

Status

approved