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%I M2736 N1098 #34 Dec 20 2021 20:28:12
%S 1,3,8,19,42,88,176,339,633,1150,2040,3544,6042,10128,16720,27219,
%T 43746,69483,109160,169758,261504,399272,604560,908248,1354427,
%U 2005710,2950544,4313232,6267642,9055856,13013440,18603603,26463168,37464230
%N Expansion of (1/theta_4(q)^2 -1)/4 in powers of q.
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002318/b002318.txt">Table of n, a(n) for n = 1..1000</a>
%F Expansion of (eta(q^2)^2 / eta(q)^4 - 1) / 4 in powers of q.
%F a(n) = A001934(n) / 4.
%e 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 + ...
%p 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)
%t Rest[CoefficientList[ Series[(1/EllipticTheta[4, 0, q]^2 - 1)/4, {q, 0, 34}], q]] (* _Jean-François Alcover_, Jul 18 2011 *)
%t 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 *)
%o (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 */
%Y Cf. A001934.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
%E More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 18 2005
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