A258100 Expansion of c(q) * c(q^3) / c(q^2)^2 in powers of q where c() is a cubic AGM theta function.

1, 1, 0, -1, -2, 0, 4, 5, 0, -10, -12, 0, 20, 26, 0, -39, -50, 0, 76, 92, 0, -140, -168, 0, 244, 295, 0, -415, -496, 0, 696, 818, 0, -1140, -1332, 0, 1820, 2126, 0, -2861, -3324, 0, 4448, 5126, 0, -6816, -7824, 0, 10292, 11793, 0, -15372, -17548, 0, 22756

Offset

0,5

Comments

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (psi(q) * f(-q^9)^3) / (chi(-q^3)^2 * psi(q^3)^4) in powers of q where psi(), chi(), f() are Ramanujan theta functions.

Expansion of eta(q^2)^2 * eta(q^3)^2 * eta(q^9)^3 / (eta(q) * eta(q^6)^6) in powers of q.

Euler transform of period 18 sequence [ 1, -1, -1, -1, 1, 3, 1, -1, -4, -1, 1, 3, 1, -1, -1, -1, 1, 0, ...].

a(n) = (-1)^n * A164616(n). a(3*n) = A128641(n). a(3*n + 1) = A258099(n). a(3*n + 2) = 0.

Convolution invserse is A182034.

Example

G.f. = 1 + q - q^3 - 2*q^4 + 4*q^6 + 5*q^7 - 10*q^9 - 12*q^10 + 20*q^12 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q^9]^3 EllipticTheta[ 2, 0, q^(1/2)] QPochhammer[ q^3]^2 / (2 q^(1/8) QPochhammer[ q^6]^6), {q, 0, n}];
a[ n_] := SeriesCoefficient[ 4 q QPochhammer[ q^9]^3 EllipticTheta[ 2, 0, q^(1/2)] / (QPochhammer[ q^3] EllipticTheta[ 2, 0, q^(3/2)]^3), {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^9 + A)^3 / (eta(x + A) * eta(x^6 + A)^6), n))};

Crossrefs

Cf. A128641, A164616, A182034, A258099.

Sequence in context: A091684 A100050 A164616 * A173335 A335118 A201837

Adjacent sequences: A258097 A258098 A258099 * A258101 A258102 A258103

Keyword

sign

Author

Michael Somos, May 20 2015

Status

approved