A001938 Expansion of k/(4*q^(1/2)) in powers of q, where k defined by sqrt(k) = theta_2(0, q)/theta_3(0, q).
1, -4, 14, -40, 101, -236, 518, -1080, 2162, -4180, 7840, -14328, 25591, -44776, 76918, -129952, 216240, -354864, 574958, -920600, 1457946, -2285452, 3548550, -5460592, 8332425, -12614088, 18953310, -28276968, 41904208, -61702876, 90304598, -131399624
Offset: 0
Examples
G.f. = 1 - 4*x + 14*x^2 - 40*x^3 + 101*x^4 - 236*x^5 + 518*x^6 - 1080*x^7 + ... G.f. of B(q) = q * A(q^2): q - 4*q^3 + 14*q^5 - 40*q^7 + 101*q^9 - 236*q^11 + 518*q^13 - 1080*q^15 + ...
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.
- E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 385.
- N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
- 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
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
- 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]
- R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Crossrefs
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ -x, x^2] QPochhammer[ x^2, x^4])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *) a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ -x])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *) a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - (-x)^k, {k, n}])^4, {x, 0, n}]; (* Michael Somos, Sep 24 2011 *) a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)] / (2 EllipticTheta[ 3, 0, q]))^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *) a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^4, {q, 0, n + 1/2}]; (* Michael Somos, Sep 24 2011 *) -
PARI
{a(n) = my(A, A2, m); if( n<0, 0, n = 2*n + 1; A = x + O(x^3); m=2; while( m -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^4, n))}; /* Michael Somos, Mar 26 2004 */
Formula
Expansion of (psi(x^2) / phi(x))^2 = (psi(x) / phi(x))^4 = (psi(x^2) / psi(x))^4 = (psi(-x) / psi(-x^2))^4 = (chi(-x) / chi(-x^2)^2)^4 = (chi(x)^2 * chi(-x))^-4 = (chi(x) * chi(-x^2))^-4 = (f(-x^4) / f(x))^4 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Feb 26 2012
G.f. A(x) satisfies 1 = (1 - 16 * x * A(x)^2) * (1 + 16 * x * A(-x)^2). - Michael Somos, Mar 26 2004
Expansion of q^(-1/2) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^4 in powers of q.
Euler transform of period 4 sequence [ -4, 8, -4, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v - (u * (1 + 4*v))^2. - Michael Somos, Mar 26 2004
G.f. A(q) satisfies A(q) = sqrt(A(q^2)) / (1 + 4*q*A(q^2)); together with limit_{n->infinity} A(x^n) = 1 this gives a fast algorithm to compute the series. - Joerg Arndt, Aug 06 2011
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)))^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139820. - Michael Somos, Jun 04 2015
G.f.: ((Sum_{n >= 0} x^(n*(n+1))) / (1 + Sum_{n >= 1} x^(n^2)))^4 (from the sum representation of the Jacobi theta functions evaluated at vanishing argument). - Wolfdieter Lang, Jul 04 2016
a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi) / (32 * 2^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
Extensions
Edited by N. J. A. Sloane, Mar 31 2007
Comments