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%I #27 Aug 24 2023 13:42:26
%S 1,0,0,4,-240,0,0,26760,-85995,0,0,1707264,-4096240,0,0,44330496,
%T -91951146,0,0,708938760,-1343913984,0,0,8277534720,-14733025125,0,0,
%U 77092288000,-130880766192,0,0,604139268096,-988226335125,0,0,4125992712192,-6548115718144,0,0,25168873498760
%N Expansion of the modular form of level 4 and weight 1/2.
%H R. E. Borcherds, <a href="http://www.math.berkeley.edu/~reb/papers/">Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras</a>, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.
%H J. H. Bruinier, <a href="https://arxiv.org/abs/math/0404427">Infinite products in number theory and geometry</a>, arXiv:math/0404427 [math.NT], 2004.
%H K. Ono, <a href="http://www.ams.org/notices/201011/rtx101101410p.pdf">The last words of a genius</a>, Notices Amer. Math. Soc., 57 (2010), 1410-1419.
%F 60*theta_3(z)+KZ(z)*E_6(4z)/del_12(4z) where KZ(z) is the cusp form of weight 13/2 given by the sequence A054891. - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jan 11 2001
%F a(4*n + 2) = a(4*n + 3) = 0. - _Michael Somos_, Feb 22 2015
%e G.f. = 1/q^3 + 4 - 240*q + 26760*q^4 - 85995*q^5 + 1707264*q^8 - 4096240*q^9 + ...
%o (PARI) {a(n) = my(A, F, t, T, E); if( n<-3, 0, n += 4; A = x * O(x^n); t = sum( k= 1, sqrtint( n), 2 * x^k^2, 1 + A); T = t^20; E = sum( k= 1, n\4, -264 * sigma( k, 9) * x^(4*k), 1 + A); polcoeff( (( E / T )' * T / eta(x^4 + A)^24 + 1056*x^3) * -1/40 * t, n-1))}; /* _Michael Somos_, Jul 08 2011 */
%Y Cf. A000025, A192732.
%K sign
%O -3,4
%A _N. J. A. Sloane_.
%E More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jan 11 2001
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