A225915 Expansion of (k(q) / 4)^4 in powers of q where k() is a Jacobi elliptic function.

1, -16, 152, -1088, 6444, -33184, 153152, -646528, 2533070, -9311664, 32387616, -107299904, 340436664, -1039026144, 3061896704, -8739810688, 24229115109, -65390485328, 172155210320, -442928464640, 1115433685796, -2753362613984, 6670224790272, -15876957230848

Offset

2,2

Links

G. C. Greubel, Table of n, a(n) for n = 2..2500

Formula

Expansion of (eta(q) * eta(q^4)^2 / eta(q^2)^3)^16 in powers of q.

Euler transform of period 4 sequence [-16, 32, -16, 0, ...].

G.f.: q^2 * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^16.

Convolution square of A005798.

a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n)) / (65536 * 2^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 17 2017

Example

G.f. = q^2 - 16*q^3 + 152*q^4 - 1088*q^5 + 6444*q^6 - 33184*q^7 + 153152*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (InverseEllipticNomeQ[  q] / 16)^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^16, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q ( Product[ 1 - q^k, {k, 4, n - 1, 4}]/
Product[ 1 - (-q)^k, {k, n - 1}])^16, {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^16, n))};

Crossrefs

Cf. A001938, A005798, A079006, A083365.

Sequence in context: A049351 A240424 A341391 * A023014 A073384 A022644

Adjacent sequences: A225912 A225913 A225914 * A225916 A225917 A225918

Keyword

sign

Author

Michael Somos, May 20 2013

Status

approved