OFFSET
-1,2
COMMENTS
REFERENCES
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
LINKS
Seiichi Manyama, Table of n, a(n) for n = -1..10000
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (b(q) * c(q^2)^3) / (c(q) * c(q^4)^2 * b(q^4)) in powers of q where b(), c() are cubic AGM functions.
Expansion of (1/q) * chi(q) * chi(-q)^5 * chi(q^3)^5 * chi(-q^3) in powers of q where chi() is a Ramanujan theta function.
Expansion of (eta(q)^4 * eta(q^6)^9) / (eta(q^2)^3 * eta(q^3)^4 * eta(q^4) * eta(q^12)^5) in powers of q.
Euler transform of period 12 sequence [ -4, -1, 0, 0, -4, -6, -4, 0, 0, -1, -4, 0, ...].
a(2*n) = 0 unless n=0. a(2*n - 1) = A058491(n).
a(n) = A187045(n) unless n=0. - Michael Somos, Sep 05 2015
EXAMPLE
G.f. = 1/q - 4 + 5*q - 5*q^3 + 9*q^5 - 14*q^7 + 19*q^9 - 34*q^11 + 55*q^13 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q, q^2] QPochhammer[ -q^3, q^6])^5 (QPochhammer[ -q, q^2] QPochhammer[ q^3, q^6]), {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A)^4 * eta(x^6 + A)^9) / (eta(x^2 + A)^3 * eta(x^3 + A)^4 * eta(x^4 + A) * eta(x^12 + A)^5), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Mar 07 2011
STATUS
approved