A298209 Expansion of (1/q) * phi(q) * phi(-q^5) / (f(-q^4) * f(-q^20)) in powers of q where phi(), f() are Ramanujan theta functions.

1, 2, 0, 0, 3, 0, -4, 0, 4, 0, -4, 0, 7, 0, -12, 0, 13, 0, -16, 0, 22, 0, -28, 0, 38, 0, -44, 0, 55, 0, -72, 0, 83, 0, -104, 0, 129, 0, -156, 0, 187, 0, -220, 0, 273, 0, -328, 0, 384, 0, -452, 0, 539, 0, -652, 0, 757, 0, -880, 0, 1041, 0, -1220, 0, 1428, 0

Offset

-1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = -1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (1/q) * chi(-q) * chi(q)^3 * chi(-q^5)^3 * chi(q^5) in powers of q where chi() is a Ramanujan theta function.

Expansion of eta(q^2)^5 * eta(q^5)^2 / (eta(q)^2 * eta(q^4)^3 * eta(q^10) * eta(q^20)) in powers of q.

Euler transform of period 20 sequence [2, -3, 2, 0, 0, -3, 2, 0, 2, -4, 2, 0, 2, -3, 0, 0, 2, -3, 2, 0, ...].

a(2*n) = 0 except n=0. a(2*n + 1) = A058559(n) for all n in Z. a(n) = -(-1)^n * A298203(n).

Example

G.f. = q^-1 + 2 + 3*q^3 - 4*q^5 + 4*q^7 - 4*q^9 + 7*q^11 - 12*q^13 + 13*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ 1/q EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^5] / (QPochhammer[ q^4] QPochhammer[ q^20]), {q, 0, n}];
a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q, q^2] QPochhammer[ -q, q^2]^3 QPochhammer[ q^5, q^10]^3 QPochhammer[ -q^5, q^10], {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^5 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^10 + A) * eta(x^20 + A)), n))};

Crossrefs

Cf. A058559, A298203.

Sequence in context: A292250 A245492 A298203 * A211871 A194586 A288437

Adjacent sequences: A298206 A298207 A298208 * A298210 A298211 A298212

Keyword

sign

Author

Michael Somos, Jan 15 2018

Status

approved