A006709 Expansion of a modular function. (Formerly M0600)

1, -2, -3, -4, 22, 30, -12, -128, -147, 132, 548, 516, -552, -1924, -1572, 1784, 5790, 4410, -5180, -15608, -11406, 13712, 39128, 27528, -33518, -92682, -63156, 77284, 208636, 139026, -170272, -449904, -294741, 360872, 936836, 604440, -739228, -1892636

Offset

-2,2

References

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

Links

G. C. Greubel, Table of n, a(n) for n = -2..1000

Morris Newman, Construction and application of a class of modular functions. II. Proc. London Math. Soc. (3) 9 1959 373-387. MR0107629 (21 #6354).

Morris Newman, Construction and application of a class of modular functions, II, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]

Formula

Expansion of ((1/q) * f(-q, -q^2)^2 * f(q, q^2) / psi(q^3)^3)^2 in powers of q where psi(), f(,) are Ramanujan theta functions. - Michael Somos, Jan 18 2015

Expansion of (eta(q) * eta(q^2))^2 * (eta(q^3)^10 / eta(q^6)^14) in powers of q. - Michael Somos, Sep 22 2005

Euler transform of period 6 sequence [ -2, -4, -12, -4, -2, 0, ...]. - Michael Somos, Sep 22 2005

G.f.: (x^-2) * (Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k))^2 * (1 - x^(3*k))^10 / (1 - x^(6*k))^14).

Example

G.f. = 1/q^2 - 2/q - 3 - 4*q + 22*q^2 + 30*q^3 - 12*q^4 - 128*q^5 +...

Mathematica

a[ n_] := SeriesCoefficient[ ( (1/q) QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^3]^5 / QPochhammer[ q^6]^7)^2, {q, 0, n}]; (* Michael Somos, Jan 18 2015 *)
a[ n_] := SeriesCoefficient[ ( EllipticTheta[ 4, 0, q^3]^2 EllipticTheta[ 4, 0, q^9] (2 / (EllipticTheta[ 4, 0, q^9] - EllipticTheta[ 4, 0, q]))^3)^2, {q, 0, 3 n}]; (* Michael Somos, Jan 18 2015 *)

Prog

(PARI) {a(n) = local(A); if ( n<-2, 0, n+=2; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A)^2 * eta(x^3 + A)^10 / eta(x^6 + A)^14, n))}; /* Michael Somos, Sep 22 2005 */
(PARI) q='q+O('q^99); Vec((eta(q)*eta(q^2))^2*(eta(q^3)^10/eta(q^6)^14)) \\ Altug Alkan, Apr 03 2018

Crossrefs

Sequence in context: A069801 A019074 A019075 * A115884 A331603 A171728

Adjacent sequences: A006706 A006707 A006708 * A006710 A006711 A006712

Keyword

sign,easy

Author

N. J. A. Sloane

Status

approved