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q-expansion of modular form of weight 13/2: eta(8 tau)^12 * theta(tau).
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%I #21 Nov 24 2015 03:39:07

%S 0,0,0,0,1,2,0,0,2,0,0,0,-12,-22,0,0,-24,0,0,0,56,84,0,0,108,0,0,0,

%T -112,-66,0,0,-176,0,0,0,9,-398,0,0,-196,0,0,0,364,990,0,0,1056,0,0,0,

%U -616,70,0,0,-728,0,0,0,432,-2354,0,0,-1472,0,0,0,-240,1080,0,0,990,0,0,0

%N q-expansion of modular form of weight 13/2: eta(8 tau)^12 * theta(tau).

%H T. D. Noe, <a href="/A002284/b002284.txt">Table of n, a(n) for n=0..1004</a>

%H R. E. Borcherds, <a href="http://www.math.berkeley.edu/~reb/papers/siegel/cuspcoefs">A Siegel cusp form of weight 12 ...</a>

%F Expansion of eta(q^2)^5 * eta(q^8)^12 / (eta(q) * eta(q^4))^2 in powers of q.

%F Euler transform of period 8 sequence [ 2, -3, 2, -1, 2, -3, 2, -13, ...]. - _Michael Somos_, Mar 06 2004

%F G.f.: x^4 * (Product_{k>0} (1 - x^(2*k))^5 * (1 - x^(8*k))^12 / ((1 - x^k) * (1 - x^(4*k)))^2). - _Michael Somos_, Mar 06 2004

%e G.f. = q^4 + 2*q^5 + 2*q^9 - 12*q^13 - 22*q^14 - 24*q^17 + 56*q^21 + ...

%t max = 76; f[x_] := x^4*Product[ (1 - x^(2k))^5 (1 - x^(8k))^12/((1 - x^k) (1 - x^(4k)))^2, {k, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* _Jean-François Alcover_, Dec 12 2011, after _Michael Somos_ *)

%t QP = QPochhammer; s = q^4*QP[q^2]^5*(QP[q^8]^12/(QP[q]*QP[q^4])^2) + O[q]^80; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 24 2015 *)

%o (PARI) {a(n) = if( n<4, 0, n-=4; polcoeff( eta(x^8 + x * O(x^n))^12 * sum(k=1, sqrtint(n), 2*x^k^2, 1), n))}; /* _Michael Somos_, Mar 06 2004 */

%o (PARI) {a(n) = local(A); if( n<4, 0, n-=4; A = x * O(x^n); polcoeff(eta(x^2 + A)^5 * eta(x^8 + A)^12 / (eta(x + A)^2 * eta(x^4 + A)^2), n))}; /* _Michael Somos_, Mar 06 2004 */

%K sign,nice

%O 0,6

%A _N. J. A. Sloane_