A226235 Expansion of q * (chi(-q) / chi(-q^3))^12 in powers of q where chi() is a Ramanujan theta function.

1, -12, 66, -220, 495, -804, 1068, -1596, 3279, -6952, 12276, -17844, 23653, -34080, 57168, -98428, 154332, -215724, 285388, -395784, 600459, -931888, 1365696, -1853076, 2426189, -3277896, 4689534, -6815008, 9538632, -12664440, 16403188, -21690876, 29812932

Offset

1,2

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Richard Moy, Congruences among power series coefficients of modular forms, arXiv:1309.4320 [math.NT], 2013.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Wenzhe Yang, Apéry's irrationality proof, mirror symmetry and Beukers' modular forms, arXiv:1911.02608 [math.NT], 2019.

Formula

Expansion of q * (f(-q, -q^5) / f(-q^6))^12 in powers of q where f() is a Ramanujan theta function.

Expansion of ((c(q^2) * b(q)) / (c(q) * b(q^2)))^3 in powers of q where b() and c() are cubic AGM theta functions.

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

Euler transform of period 6 sequence [ -12, 0, 0, 0, -12, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w)= (v - u^2) * (v - w^2) - u*w * (24*(1 + v^2) + 152*v).

G.f. A(x) satisfies f(x) = g(A(x)) where f, g are the g.f. for A006353 and A005259.

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = f(t) where q = exp(2 Pi i t).

G.f.: x * (Product_{k>0} 1 - x^k + x^(2*k))^12 where 1 - x + x^2 is the 6th cyclotomic polynomial.

Convolution inverse of A121665. Convolution 12th power of A109389.

a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n/3)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 30 2017

Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 11 + 5*sqrt(3) - sqrt(189 + 114*sqrt(3)). - Simon Plouffe, Mar 02 2021

Example

G.f. = q - 12*q^2 + 66*q^3 - 220*q^4 + 495*q^5 - 804*q^6 + 1068*q^7 - 1596*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^6] / (QPochhammer[ q^2] QPochhammer[ q^3]))^12, {q, 0, n}]
nmax = 50; CoefficientList[Series[Product[((1 + x^(3*k))/(1 + x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A)))^12, n))}

Crossrefs

Cf. A005259, A006353, A109389, A121665.

Sequence in context: A010928 A080559 A284641 * A045853 A277104 A014787

Adjacent sequences: A226232 A226233 A226234 * A226236 A226237 A226238

Keyword

sign

Author

Michael Somos, Sep 18 2013

Status

approved