A187053 Expansion of (psi(x^2) / psi(x))^3 in powers of x where psi() is a Ramanujan theta function.

1, -3, 9, -22, 48, -99, 194, -363, 657, -1155, 1977, -3312, 5443, -8787, 13968, -21894, 33873, -51795, 78345, -117312, 174033, -255945, 373353, -540486, 776848, -1109040, 1573209, -2218198, 3109713, -4335840, 6014123, -8300811, 11402928

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.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)

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]

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-3/8) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^3 in powers of q.

Euler transform of period 4 sequence [-3, 6, -3, 0, ...].

G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k-1)))^3.

Convolution inverse of A029840. Convolution cube of A083365. a(n) = (-1)^n * A001937(n).

a(n) ~ (-1)^n * 3^(1/4) * exp(sqrt(3*n/2)*Pi) / (16*2^(3/4)*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

Example

G.f. = 1 - 3*x + 9*x^2 - 22*x^3 + 48*x^4 - 99*x^5 + 194*x^6 - 363*x^7 + ...
G.f. = q^3 - 3*q^11 + 9*q^19 - 22*q^27 + 48*q^35 - 99*q^43 + 194*q^51 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ -x])^3, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

Prog

(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)^3, n))};

Crossrefs

Cf. A001937, A029840, A083365.

Sequence in context: A217879 A217878 A192389 * A001937 A086817 A247188

Adjacent sequences: A187050 A187051 A187052 * A187054 A187055 A187056

Keyword

sign

Author

Michael Somos, Mar 06 2011

Status

approved