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%I M3332 #70 Aug 07 2023 14:58:14
%S 1,4,8,16,32,56,96,160,256,404,624,944,1408,2072,3008,4320,6144,8648,
%T 12072,16720,22976,31360,42528,57312,76800,102364,135728,179104,
%U 235264,307672,400704,519808,671744,864960,1109904,1419456,1809568,2299832
%N Expansion of theta_3 / theta_4.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Number of partitions of 2n into parts with 2 types c, c* of each part. The even parts appears with multiplicity 1 for each type. The odd parts appears with multiplicity 2 (cc or c*c* but not cc*, that is, no mixing is allowed). E.g., a(4)=8 because of 44*, 22*, 211, 21*1*, 2*1*1*, 2*11, 111*1*. - _Noureddine Chair_, Jan 27 2005
%C a(n) is the number of pairs of overpartitions into odd parts where the sum of all parts is equal to n. - _Jeremy Lovejoy_, Aug 29 2020
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A007096/b007096.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 11.
%H Bernard L.S. Lin, <a href="https://doi.org/10.37236/2274">Arithmetic properties of overpartition pairs into odd parts</a>, Electronic J. Combin. 19, 2012, Paper 17.
%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 [4, -2, 4, 0, ...]. - _Vladeta Jovovic_, Mar 22 2005
%F Expansion of eta(q^2)^6 /(eta(q)^4 * eta(q^4)^2) in powers of q.
%F Expansion of phi(q) / phi(-q) = chi(q)^2 / chi(-q)^2 = psi(q)^2 / psi(-q)^2 = phi(-q^2)^2 / phi(-q)^2 = phi(q)^2 / phi(-q^2)^2 = chi(-q^2)^2 / chi(-q)^4 = chi(q)^4 / chi(-q^2)^2 = f(q)^2 / f(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4. - _Michael Somos_, Jan 01 2006
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A028939.
%F Expansion of Jacobian elliptic function 1 / sqrt(k') in powers of q. - see Fine.
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + u^2 - 2*u*v^2. - _Michael Somos_, Jul 07 2005
%F Unique solution to f(x^2)^2 = (f(x) + 1 / f(x)) / 2 and f(0)=1, f'(0) nonzero.
%F G.f.: theta_3 / theta_4 = (Sum_{k} x^k^2) / (Sum_{k} (-x)^k^2) = (Product_{k>0} (1 - x^(4*k - 2)) / ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2)^2.
%F A097243(n) = a(4*n). 8*A022577(n) = a(4*n + 2). a(n) = 4*A123655(n) if n>0. Convolution square of A080054.
%F Empirical: sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(1/4). - _Simon Plouffe_, Feb 20 2011
%F Empirical : sum(exp(-Pi*sqrt(2))^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = (-2+2*2^(1/2))^(1/4). - _Simon Plouffe_, Feb 20 2011
%F Empirical : sum(exp(-2*Pi)^(n-1)*a(n),n=1..infinity) = 1/2*(8+6*2^(1/2))^(1/4). - _Simon Plouffe_, Feb 20 2011
%F a(n) ~ exp(Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - _Vaclav Kotesovec_, Aug 28 2015
%F G.f.: exp(4*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - _Ilya Gutkovskiy_, Apr 19 2019
%e G.f. = 1 + 4*q + 8*q^2 + 16*q^3 + 32*q^4 + 56*q^5 + 96*q^6 + 160*q^7 + 256*q^8 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(-1/4), {q, 0, n}]]; (* _Michael Somos_, Jul 11 2011 *)
%t a[ n_] := SeriesCoefficient[( QPochhammer[ -q, q^2] / QPochhammer[ q, q^2])^2, {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t a[ n_] := SeriesCoefficient[ (Product[ 1 - (-q)^k, {k, n}] / Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 28 2015 *)
%o (PARI) {a(n) = my(A, B); if( n<0, 0, A = 1 + 4*x; for( k=2, n, B = A + x^2 * O(x^k); A += Pol(2 * subst(B, x, x^2)^2 - B - 1/B) / x / 8); polcoeff(A, n))}; /* _Michael Somos_, Jul 07 2005*/
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)))^2, n))}; /* _Michael Somos_, Jan 01 2006 */
%Y Self-convolution of A080054. - _Vladeta Jovovic_, Mar 22 2005
%Y Cf. A014969, A001936, A001938, A079006, A127391, A127392.
%Y Cf. A022577, A080054, A097243, A123655.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
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