A002318 Expansion of (1/theta_4(q)^2 -1)/4 in powers of q.

1, 3, 8, 19, 42, 88, 176, 339, 633, 1150, 2040, 3544, 6042, 10128, 16720, 27219, 43746, 69483, 109160, 169758, 261504, 399272, 604560, 908248, 1354427, 2005710, 2950544, 4313232, 6267642, 9055856, 13013440, 18603603, 26463168, 37464230

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			q + 3*q^2 + 8*q^3 + 19*q^4 + 42*q^5 + 88*q^6 + 176*q^7 + 339*q^8 + 633*q^9 + ...
		

References

• J. W. L. Glaisher, "On the Coefficients in the q-series for pi/2K and 2G/pi", Quart J. Pure and Applied Math., 21 (1885), 60-76.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001934.

Programs

Maple

seq(coeff(convert(series(mul(( 1 - x^k )^(-(2+(k mod 2)*2)),k=1..100),x,100),polynom),x,i)/4,i=1..50); (Pab Ter)

Mathematica

Rest[CoefficientList[ Series[(1/EllipticTheta[4, 0, q]^2 - 1)/4, {q, 0, 34}], q]] (* Jean-François Alcover, Jul 18 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Integrate[ (EllipticK[m] - EllipticE[m]) / (8 Sqrt[1 - m] (Pi/2) q), q], {q, 0, n}]] (* Michael Somos, Jan 24 2012 *)

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A)^4 - 1, n) / 4)} /* Michael Somos, Feb 09 2006 */

Formula

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

a(n) = A001934(n) / 4.

Extensions

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 18 2005