cp's OEIS Frontend

A248394 q-Expansion of the modular form of weight 3/2, g*theta(2) in Tunnell's notation (see Comments).

%I A248394 #53 Apr 10 2025 23:21:22
%S A248394 0,1,0,2,0,0,0,0,0,1,0,-2,0,0,0,0,0,-4,0,-2,0,0,0,0,0,-3,0,0,0,0,0,0,
%T A248394 0,4,0,-4,0,0,0,0,0,0,0,6,0,0,0,0,0,1,0,4,0,0,0,0,0,4,0,2,0,0,0,0,0,0,
%U A248394 0,-2,0,0,0,0,0,4,0,-2
%N A248394 q-Expansion of the modular form of weight 3/2, g*theta(2) in Tunnell's notation (see Comments).
%C A248394 g = q*Product_{m=1..oo} (1-q^(8*m))*(1-q^(16*m)),
%C A248394 theta(t) = Sum_{n=-oo..oo} q^(t*n^2).
%C A248394 Although the OEIS does not normally include sequences in which every other term is zero, this one is important enough to warrant an exception.
%H A248394 Seiichi Manyama, <a href="/A248394/b248394.txt">Table of n, a(n) for n = 0..10000</a>
%H A248394 J. B. Tunnell, <a href="http://dx.doi.org/10.1007/BF01389327">A classical Diophantine problem and modular forms of weight 3/2</a>, Invent. Math., 72 (1983), 323-334.
%H A248394 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A248394 From _Seiichi Manyama_, Sep 30 2018: (Start)
%F A248394 Let q = exp(Pi i t).
%F A248394 theta_3(q) = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + ... .
%F A248394 G.f.: (theta_3(q) - theta_3(q^4))*(theta_3(q^32) - theta_3(q^8)/2)*theta_3(q^2).
%F A248394 a(2*n-1) = A080918(2*n-1) - A080917(2*n-1)/2 = A072069(n) - A072068(n)/2 for n > 0. (End)
%p A248394 # This produces a list of the first 100 terms:
%p A248394 g:=q*mul((1-q^(8*m))*(1-q^(16*m)),m=1..30);
%p A248394 g:=series(g,q,100);
%p A248394 th:=t->series( add(q^(t*n^2),n=-50..50), q, 100);
%p A248394 series(g*th(2),q,100);
%p A248394 seriestolist(%);
%p A248394 # Alternative with https://oeis.org/transforms.txt and the Somos Euler transform in A034950:
%p A248394 p8 := [2,-3,2,-2,2,-3,2,-3] ;
%p A248394 L := [seq(op(p8),i=1..10)] ;
%p A248394 EULER(%) ;
%p A248394 [1,op(%)] ;
%p A248394 [0,op(AERATE(%,1))] ; # _R. J. Mathar_, Nov 11 2014
%t A248394 QP = QPochhammer; s = q*QP[q^8]*QP[q^16]*EllipticTheta[3, 0, q^2] + O[q]^80; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%Y A248394 The nonzero bisection is A034950, which has further information and references.
%Y A248394 Used in A248397-A248406.
%Y A248394 Cf. A000122 (theta_3(q)), A072068, A072069, A080917, A080918, A248395.
%K A248394 sign
%O A248394 0,4
%A A248394 _N. J. A. Sloane_, Oct 18 2014