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A185717 Expansion of q^(-1) * c(q^2) * (c(q) - c(q^4)) / 9 in powers of q^2 where c() is a cubic AGM theta function. 4
1, 3, 6, 8, 9, 12, 14, 18, 18, 20, 24, 24, 31, 27, 30, 32, 36, 48, 38, 42, 42, 44, 54, 48, 57, 54, 54, 72, 60, 60, 62, 72, 84, 68, 72, 72, 74, 93, 96, 80, 81, 84, 108, 90, 90, 112, 96, 120, 98, 108, 102, 104, 144, 108, 110, 114, 114, 144, 126, 144, 133, 126, 156, 128 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 = 0..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of phi(-x^3)^4 / (chi(-x) * chi(-x^3))^3 in powers of x where phi(), chi() are Ramanujan theta functions.

Euler transform of period 6 sequence [ 3, 0, -2, 0, 3, -4, ...].

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 3^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A118271.

a(3*n + 1) = 3 * a(n). A078708(2*n + 1) = A121443(2*n + 1) = A124449(2*n + 1).

EXAMPLE

1 + 3*x + 6*x^2 + 8*x^3 + 9*x^4 + 12*x^5 + 14*x^6 + 18*x^7 + 18*x^8 + ...

q + 3*q^3 + 6*q^5 + 8*q^7 + 9*q^9 + 12*q^11 + 14*q^13 + 18*q^15 + ...

MATHEMATICA

A185717[n_] := SeriesCoefficient[(QPochhammer[q^3, q^3]/QPochhammer[-q^3, q^3])^4*(1/(QPochhammer[q, q^2]*QPochhammer[q^3, q^6])^3), {q, 0, n}];

Table[A185717[n], {n, 0, 50}] (* G. C. Greubel, Jul 10 2017 *)

PROG

(PARI) {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, if( (n/d) % 3, 1, 0) * d))}

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^5 / (eta(x + A)^3 * eta(x^6 + A)), n))}

CROSSREFS

Cf. A078708, A118271, A121443, A124449.

Sequence in context: A275608 A155723 A095277 * A189637 A182338 A267415

Adjacent sequences:  A185714 A185715 A185716 * A185718 A185719 A185720

KEYWORD

nonn

AUTHOR

Michael Somos, Feb 10 2011

STATUS

approved

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Last modified April 5 06:13 EDT 2020. Contains 333238 sequences. (Running on oeis4.)