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%I #66 Sep 08 2022 08:44:51
%S 1,1,1,2,0,1,2,1,1,1,1,0,3,1,0,2,1,1,1,0,1,3,1,2,0,0,1,2,1,0,3,1,0,2,
%T 1,1,2,0,1,0,2,1,2,1,0,3,0,1,3,0,0,2,1,0,0,1,2,4,1,1,0,1,1,1,0,1,3,1,
%U 1,0,1,1,2,1,0,3,0,1,4,0,1,0,1,0,2,1,1,2,0,0,2,2,1,3,0,0,2,2,1,0,2,1,0,1,0
%N Product t2(q^d); d | 2, where t2 = theta2(q)/(2*q^(1/4)).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Also the number of representations of n as the sum of a triangular number and twice a triangular number. - _James A. Sellers_, Dec 21 2005
%C Also the number of positive odd solutions to equation x^2 + 2*y^2 = 8*n + 3. - _Seiichi Manyama_, May 28 2017
%H Seiichi Manyama, <a href="/A033761/b033761.txt">Table of n, a(n) for n = 0..10000</a>
%H R. P. Agarwal, <a href="https://www.ias.ac.in/article/fulltext/pmsc/103/03/0269-0293">Lambert series and Ramanujan</a>, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
%H Iana I. Anguelova, <a href="https://arxiv.org/abs/1708.04992">The two bosonizations of the CKP hierarchy: overview and character identities</a>, arXiv:1708.04992 [math-ph], 2017.
%H M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/j.disc.2004.08.045">The number of representations of a number by various forms</a>, Discrete Mathematics 298 (2005), 205-211.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Euler transform of period 4 sequence [1, 0, 1, -2, ...]. - _Vladeta Jovovic_, Sep 14 2004
%F Expansion of psi(q) * psi(q^2) in powers of q where psi() is a Ramanujan theta function.
%F Expansion of q^(-3/8) * eta(q^2) * eta^2(q^4) / eta(q) in powers of q. - _Michael Somos_, Jul 05 2006
%F Expansion of q^(-3/4) * (theta_2(q) * theta_2(q^2)) / 4 in powers of q^2. - _Michael Somos_, Jul 05 2006
%F Given g.f. A(x), then B(x) = x^3 * A(x^8) satisfies 0 = f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6) = u1^4*u6^2 + 3*u2^2*u3^4 - 4*u1*u2*u3*u6 * (u2^2 + 3*u6^2). - _Michael Somos_, Jul 05 2006
%F a(n) = A002325(8*n+3)/2. [Hirschhorn] - _R. J. Mathar_, Mar 23 2011
%F a(n) = A027414(8*n + 3). - _Michael Somos_, Nov 16 2011
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082564. - _Michael Somos_, Jan 31 2015
%F From _Peter Bala_, Jan 07 2021: (Start)
%F G.f.: A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(8*n + 3)). See Agarwal, p. 285, equation 6.19.
%F A(x^2) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(8*n + 3)). Cf. A121444. (End)
%F A(q^2) = (1/2)*Sum_{k >= 0} q^k/(1 + q^(4*k+3)) + (1/2)*Sum_{k >= 0} q^(3*k)/(1 + q^(4*k+1)) - set z = 1 and replace q with q^2 in Anguelova, equation 3.35. - _Peter Bala_, Mar 03 2021
%e G.f. = 1 + x + x^2 + 2*x^3 + x^5 + 2*x^6 + x^7 + x^8 + x^9 + x^10 + 3*x^12 + ...
%e G.f. = q^3 + q^11 + q^19 + 2*q^27 + q^43 + 2*q^51 + q^59 + q^67 + q^75 + q^83 + ...
%p sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
%p A002325 := proc(n) sigmamr(n,8,1)+sigmamr(n,8,3)-sigmamr(n,8,5)-sigmamr(n,8,7) ; end proc:
%p A033761 := proc(n) A002325(8*n+3)/2 ; end proc:
%p seq(A033761(n),n=0..90) ; # _R. J. Mathar_, Mar 23 2011
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^2] / 4, {q, 0, 2 n + 3/4}]; (* _Michael Somos_, Nov 16 2011 *)
%t QP = QPochhammer; s = QP[q^2]*(QP[q^4]^2/QP[q]) + O[q]^105; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015, adapted from PARI *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 / eta(x + A), n))}; /* _Michael Somos_, Jul 05 2006 */
%o (Magma) A := Basis( ModularForms( Gamma1(32), 1), 840); A[4] + A[12]; /* _Michael Somos_, Jan 31 2015 */
%Y Cf. A027414, A097723, A033761-A033807, A082564.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Sep 14 2004
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