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 A002318 Expansion of (1/theta_4(q)^2 -1)/4 in powers of q. (Formerly M2736 N1098) 4
 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; text; 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). 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 EXTENSIONS More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 18 2005 STATUS approved

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Last modified May 18 22:36 EDT 2021. Contains 344005 sequences. (Running on oeis4.)