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%I M3443 N1397 #95 Jul 30 2024 04:21:00
%S 1,4,12,32,76,168,352,704,1356,2532,4600,8160,14176,24168,40512,66880,
%T 108876,174984,277932,436640,679032,1046016,1597088,2418240,3632992,
%U 5417708,8022840,11802176,17252928,25070568,36223424,52053760,74414412
%N Expansion of 1/theta_4(q)^2 in powers of q.
%C Euler transform of period 2 sequence [ 4, 2, ...].
%C The Cayley reference actually is to A004403. - _Michael Somos_, Feb 24 2011
%C Number of overpartition pairs, see Lovejoy reference. - _Joerg Arndt, Apr 03 2011
%C In general, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^m and m>=1, then a(n) ~ exp(Pi*sqrt(m*n)) * m^((m+1)/4) / (2^(3*(m+1)/2) * n^((m+3)/4)). - _Vaclav Kotesovec_, Aug 17 2015
%D A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi and Vaclav Kotesovec, <a href="/A001934/b001934.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)
%H A. Cayley, <a href="/A001934/a001934.pdf">A memoir on the transformation of elliptic functions</a>, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
%H B. Kim, <a href="http://dx.doi.org/10.1016/j.disc.2011.02.002">Overpartition pairs modulo powers of 2</a>, Discrete Math., 311 (2011), 835-840.
%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. 8.
%H Jeremy Lovejoy, <a href="http://dx.doi.org/10.5802/aif.2199">Overpartition pairs</a>, Annales de l'institut Fourier, vol.56, no.3, p.781-794, 2006.
%F G.f.: Product ( 1 - x^k )^{-c(k)}, c(k) = 4, 2, 4, 2, 4, 2, ....
%F G.f.: Product{i>=1} (1+x^i)^2/(1-x^i)^2. - _Jon Perry_, Apr 04 2004
%F Expansion of eta(q^2)^2/eta(q)^4 in powers of q, where eta(x)=prod(n>=1,1-q^n).
%F a(n) = (-1)^n * A004403(n). a(n) = 4 * A002318(n) unless n=0. - _Michael Somos_, Feb 24 2011
%F a(n) ~ exp(Pi*sqrt(2*n)) / (2^(15/4) * n^(5/4)) * (1 - 15/(8*Pi*sqrt(2*n)) + 105/(256*Pi^2*n)). - _Vaclav Kotesovec_, Aug 17 2015, extended Jan 22 2017
%F a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - _Seiichi Manyama_, May 02 2017
%F G.f.: exp(2*Sum_{k>=1} (sigma(2*k) - sigma(k))*x^k/k). - _Ilya Gutkovskiy_, Sep 19 2018
%F The g.f. A(q^2) = 1/(F(q)*F(-q)), where F(q) = theta_3(q) = Sum_{n = -oo..oo} q^(n^2) is the g.f. of A000122. Cf. A002513. - _Peter Bala_, Sep 26 2023
%p mul((1+x^n)^2/(1-x^n)^2,n=1..256);
%t CoefficientList[Series[1/EllipticTheta[4, 0, q]^2, {q, 0, 32}], q] (* _Jean-François Alcover_, Jul 18 2011 *)
%t nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 17 2015 *)
%t QP = QPochhammer; s = QP[q^2]^2/QP[q]^4 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Dec 01 2015, adapted from PARI *)
%o (PARI) my(N=33, x='x+O('x^N)); Vec(prod(i=1, N, (1+x^i)^2/(1-x^i)^2))
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A)^4, n))} /* _Michael Somos_, Feb 09 2006 */
%o (Julia) # JacobiTheta4 is defined in A002448.
%o A001934List(len) = JacobiTheta4(len, -2)
%o A001934List(33) |> println # _Peter Luschny_, Mar 12 2018
%Y Cf. A000122, A000203, A004403, A002318, A002513, A015128, A004404-A004425.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
%E More terms from _James A. Sellers_, Sep 08 2000
%E Edited by _N. J. A. Sloane_, May 13 2008 to remove an incorrect g.f.
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