A002318 Expansion of (1/theta_4(q)^2 -1)/4 in powers of q. (Formerly M2736 N1098)

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

1,2

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

Formula

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

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

Example

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 + ...

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 *)

Prog

(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 */

Crossrefs

Cf. A001934.

Sequence in context: A298406 A074839 A262156 * A229198 A095681 A079583

Adjacent sequences: A002315 A002316 A002317 * A002319 A002320 A002321

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved