A253185 Expansion of (phi(-q) * phi(-q^23))^2 in powers of q where phi() is a Ramanujan theta function.

1, -4, 4, 0, 4, -8, 0, 0, 4, -4, 8, 0, 0, -8, 0, 0, 4, -8, 4, 0, 8, 0, 0, -4, 16, -28, 8, -16, 32, -8, 0, -16, 20, -32, 8, 0, 36, -8, 0, -16, 40, -24, 0, -32, 0, -8, 4, -16, 64, -36, 28, -32, 40, -8, 16, -32, 32, -32, 8, -48, 32, -8, 16, -64, 52, -16, 32, 0

Offset

0,2

Comments

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

a(n) differs from A104794(n) = (-1)^n*A004018(n) from a(23) = -4 on. - M. F. Hasler, Mar 08 2018

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 (eta(q) * eta(q^23))^4 / (eta(q^2) * eta(q^46))^2 in powers of q.

Euler transform of a period 46 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (46 t)) = 368 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A253183.

G.f.: (Sum_{k in Z} q^k^2)^2 * (Sum_{k in Z} q^(23*k^2))^2.

Example

G.f. = 1 - 4*q + 4*q^2 + 4*q^4 - 8*q^5 + 4*q^8 - 4*q^9 + 8*q^10 - 8*q^13 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^23])^4 / (QPochhammer[ q^2] QPochhammer[ q^46])^2, {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^23 + A))^4 / (eta(x^2 + A) * eta(x^46 + A))^2, n))};
(PARI) {A253185(n, o=O('x^(n+1)))= polcoeff(((eta('x+o)*eta('x^23+o))^2/(eta('x^2+o)*eta('x^46+o)))^2, n)} \\ Writing the g.f. as a square makes the code more than 2 x faster. Using 'x prevents erroneous results in case x is used elsewhere. - M. F. Hasler, Mar 08 2018
(PARI) A253185_vec(N)={my(q='q+O('q^N)); Vec((eta(q) * eta(q^23))^4 / (eta(q^2) * eta(q^46))^2)} \\ Joerg Arndt, Mar 09 2018

Crossrefs

Cf. A104794, A253183.

Sequence in context: A164613 A104794 A004018 * A028658 A241535 A169784

Adjacent sequences: A253182 A253183 A253184 * A253186 A253187 A253188

Keyword

sign,changed

Author

Michael Somos, Mar 24 2015

Status

approved