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A001938 Expansion of k/(4*q^(1/2)) in powers of q, where k is the elliptic function defined by sqrt(k) = theta_2/theta_3.
(Formerly M3475 N1412)
16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

T. D. Noe, Table of n, a(n) for n=0..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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(x) = x * A(x^2) satisfies 0 = f(B(x), B(x^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.

a(n) = (-1)^n * A093160(n). Convolution square of A079006.

EXAMPLE

1 - 4*x + 14*x^2 - 40*x^3 + 101*x^4 - 236*x^5 + 518*x^6 - 1080*x^7 + ...

q - 4*q^3 + 14*q^5 - 40*q^7 + 101*q^9 - 236*q^11 + 518*q^13 - 1080*q^15 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ -q, q^2] QPochhammer[ q^2, q^4])^4, {q, 0, n}] (* Michael Somos, Sep 24 2011 *)

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^4, q^4] / QPochhammer[ -q, -q])^4, {q, 0, n}] (* Michael Somos, Sep 24 2011 *)

a[ n_] := SeriesCoefficient[ (Product[ 1 - q^k, {k, 4, n, 4}] / Product[ 1 - (-q)^k, {k, n}])^4, {q, 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 *)

PROG

(PARI) {a(n) = local(A, A2, m); if( n<0, 0, n = 2*n + 1; A = x + O(x^3); m=2; while( m<n, m*=2; A = subst(A, x, x^2); A = sqrt(A) / (1 + 4*A)); polcoeff(A, n))} /* Michael Somos, Mar 26 2004 */

(PARI) {a(n) = local(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 */

CROSSREFS

Cf. A001936, A079006, A093160, A127931, A127932, A127393.

Sequence in context: A187594 A066375 A093160 * A066368 A160463 A121593

Adjacent sequences:  A001935 A001936 A001937 * A001939 A001940 A001941

KEYWORD

sign,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Edited by N. J. A. Sloane, Mar 31 2007

STATUS

approved

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Last modified October 21 08:07 EDT 2014. Contains 248375 sequences.