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A230446 Expansion of q^(-1) * f(q) * f(q^7) / (f(-q^4) * f(-q^28)) in powers of q where f() is a Ramanujan theta function. 2
1, 1, -1, 0, 1, 0, -1, 0, 3, 0, -2, 0, 2, 0, -5, 0, 6, 0, -7, 0, 7, 0, -9, 0, 12, 0, -13, 0, 16, 0, -20, 0, 25, 0, -27, 0, 31, 0, -38, 0, 44, 0, -51, 0, 58, 0, -69, 0, 80, 0, -92, 0, 102, 0, -118, 0, 141, 0, -157, 0, 177, 0, -203, 0, 234, 0, -261, 0, 292, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,9

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..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-1) * chi(q) * chi(-q^2) * chi(q^7) * chi(-q^14) in power of q where chi() is a Ramanujan theta function.

Expansion of (eta(q^2) * eta(q^14))^3 / (eta(q) * eta(q^4)^2 * eta(q^7) * eta(q^28)^2) in powers of q.

Euler transform of period 28 sequence [ 1, -2, 1, 0, 1, -2, 2, 0, 1, -2, 1, 0, 1, -4, 1, 0, 1, -2, 1, 0, 2, -2, 1, 0, 1, -2, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - 2)^2 - u * v * (u - 2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A123862.

a(n) = A161970(n) unless n=0. a(n) = -(-1)^n * A161970(n). a(2*n) = 0 unless n=0.

EXAMPLE

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

MATHEMATICA

a[ n_] := SeriesCoefficient[ q^-1 QPochhammer[ -q] QPochhammer[ -q^7] / (QPochhammer[ q^4] QPochhammer[ q^28]), {q, 0, n}]

PROG

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^14 + A))^3 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^7 + A) * eta(x^28 + A)^2), n))}

CROSSREFS

Cf. A123862, A161970.

Sequence in context: A190544 A172293 A161970 * A260737 A059339 A241181

Adjacent sequences:  A230443 A230444 A230445 * A230447 A230448 A230449

KEYWORD

sign

AUTHOR

Michael Somos, Oct 18 2013

STATUS

approved

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Last modified September 27 19:34 EDT 2021. Contains 347694 sequences. (Running on oeis4.)