|
| |
|
|
A002318
|
|
Expansion of (1/theta_4(q)^2 -1)/4 in powers of q.
(Formerly M2736 N1098)
|
|
2
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
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).
|
|
|
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]] (* From 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: A178457 A072916 A074839 * A095681 A079583 A099050
Adjacent sequences: A002315 A002316 A002317 * A002319 A002320 A002321
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 18 2005
|
| |
|
|
