A001934 Expansion of 1/theta_4(q)^2 in powers of q. (Formerly M3443 N1397)

1, 4, 12, 32, 76, 168, 352, 704, 1356, 2532, 4600, 8160, 14176, 24168, 40512, 66880, 108876, 174984, 277932, 436640, 679032, 1046016, 1597088, 2418240, 3632992, 5417708, 8022840, 11802176, 17252928, 25070568, 36223424, 52053760, 74414412

Offset

0,2

Comments

Euler transform of period 2 sequence [ 4, 2, ...].

The Cayley reference actually is to A004403. - Michael Somos, Feb 24 2011

Number of overpartition pairs, see Lovejoy reference. - _Joerg Arndt, Apr 03 2011

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

References

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.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)

A. Cayley, A memoir on the transformation of elliptic functions, 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]

B. Kim, Overpartition pairs modulo powers of 2, Discrete Math., 311 (2011), 835-840.

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.

Jeremy Lovejoy, Overpartition pairs, Annales de l'institut Fourier, vol.56, no.3, p.781-794, 2006.

Formula

G.f.: Product ( 1 - x^k )^{-c(k)}, c(k) = 4, 2, 4, 2, 4, 2, ....

G.f.: Product{i>=1} (1+x^i)^2/(1-x^i)^2. - Jon Perry, Apr 04 2004

Expansion of eta(q^2)^2/eta(q)^4 in powers of q, where eta(x)=prod(n>=1,1-q^n).

a(n) = (-1)^n * A004403(n). a(n) = 4 * A002318(n) unless n=0. - Michael Somos, Feb 24 2011

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

a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017

G.f.: exp(2*Sum_{k>=1} (sigma(2*k) - sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

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

Maple

mul((1+x^n)^2/(1-x^n)^2, n=1..256);

Mathematica

CoefficientList[Series[1/EllipticTheta[4, 0, q]^2, {q, 0, 32}], q]  (* Jean-François Alcover, Jul 18 2011 *)
nmax = 40; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
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 *)

Prog

(PARI) my(x='x+O('x^30), y=prod(i=1, 30, (1+x^i)^2/(1-x^i)^2); for(i=0, 20, print1(polcoeff(y, i), ", "))
(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 */
(Julia) # JacobiTheta4 is defined in A002448.
A001934List(len) = JacobiTheta4(len, -2)
A001934List(33) |> println # Peter Luschny, Mar 12 2018

Crossrefs

Cf. A000122, A000203, A004403, A002318, A002513, A015128, A004404-A004425.

Sequence in context: A127811 A361099 A138517 * A004403 A084566 A208903

Adjacent sequences: A001931 A001932 A001933 * A001935 A001936 A001937

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from James A. Sellers, Sep 08 2000

Edited by N. J. A. Sloane, May 13 2008 to remove an incorrect g.f.

Status

approved