A209676 Expansion of f(x)^12 in powers of x where f() is a Ramanujan theta function.

1, 12, 54, 88, -99, -540, -418, 648, 594, -836, 1056, 4104, -209, -4104, -594, -4256, -6480, 4752, -298, -5016, 17226, 12100, -5346, 1296, -9063, 7128, 19494, -29160, -10032, 7668, -34738, -8712, -22572, -21812, 49248, 46872, 67562, -2508, -47520, 76912

Offset

0,2

Comments

Number 35 of the 74 eta-quotients listed in Table I of Martin (1996). See g.f. B(q) below: cusp form weight 6 level 16.

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..2500

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^12 in powers of q.

Euler transform of period 4 sequence [ 12, -24, 12, -12, ...].

a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)) otherwise.

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 4096 (t/i)^6 f(t) where q = exp(2 Pi i t).

G.f.: (Product_{k>0} (1 - (-x)^k))^12.

a(n) = (-1)^n * A000735(n).

Convolution cube of A187076. Convolution fourth power of A133089. Convolution twelfth power of A121373.

Example

G.f. = 1 + 12*x + 54*x^2 + 88*x^3 - 99*x^4 - 540*x^5 - 418*x^6 + 648*x^7 + ...
G.f. B(q) of {b(n)}: q + 12*q^3 + 54*q^5 + 88*q^7 - 99*q^9 - 540*q^11 - 418*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^12, {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A)))^12, n))};
(Magma) A := Basis( CuspForms( Gamma0(16), 6), 81); A[1] + 12*A[3] + 54*A[5] + 88*A[7]; /* Michael Somos, Jun 09 2015 */

Crossrefs

Cf. A121373, A133089, A187076.

A000735 is the same except for signs.

Sequence in context: A133078 A034436 A186210 * A000735 A341558 A022704

Adjacent sequences: A209673 A209674 A209675 * A209677 A209678 A209679

Keyword

sign

Author

Michael Somos, Mar 11 2012

Status

approved