A227454 Expansion of q * (f(q^9) / f(q))^3 in powers of q where f() is a Ramanujan theta function.

1, -3, 9, -22, 51, -108, 221, -429, 810, -1476, 2631, -4572, 7802, -13056, 21519, -34918, 55935, -88452, 138332, -213990, 327852, -497592, 748833, -1117692, 1655719, -2434938, 3556791, -5161808, 7445631, -10677096, 15226658, -21599469, 30485268, -42817788

Offset

1,2

Comments

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

Zagier (2009) denotes the g.f. as t(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 = 1..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 c(-q^3) / (-3 * b(-q)) in powers of q where b(), c() are cubic AGM theta functions.

Expansion of (eta(q) * eta(q^4) * eta(q^18)^3 / (eta(q^2)^3 * eta(q^9) * eta(q^36)))^3 in powers of q.

Euler transform of period 36 sequence [ -3, 6, -3, 3, -3, 6, -3, 3, 0, 6, -3, 3, -3, 6, -3, 3, -3, 0, -3, 3, -3, 6, -3, 3, -3, 6, 0, 3, -3, 6, -3, 3, -3, 6, -3, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227498.

G.f. t(q) satisfies f(q) = F(t(q)) where F() is the g.f. of A006077 and f() is the g.f. of A226535

G.f.: x * (Product_{k>0} (1 - (-x)^(9*k)) / (1 - (-x)^k))^3.

a(n) = -(-1)^n * A121589(n).

Example

G.f. = q - 3*q^2 + 9*q^3 - 22*q^4 + 51*q^5 - 108*q^6 + 221*q^7 - 429*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ -q^9] / QPochhammer[ -q])^3, {q, 0, n}]

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^3 / (eta(x^2 + A)^3 * eta(x^9 + A) * eta(x^36 + A)))^3, n))}

Crossrefs

Cf. A006077, A121589, A226535, A227498.

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: A365665 A160526 A121589 * A000716 A001628 A099166

Adjacent sequences: A227451 A227452 A227453 * A227455 A227456 A227457

Keyword

sign

Author

Michael Somos, Sep 22 2013

Status

approved