OFFSET
-1,2
COMMENTS
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 chi(q^9)^3 / (q * chi(q^3)) - 2 + 4 * q * chi(q^3) / chi(q^9)^3 in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^6)^18 * eta(q^9)^3 * eta(q^36)^3 / (eta(q^2)^5 * eta(q^3)^7 * eta(q^12)^7 * eta(q^18)^9) in powers of q.
Euler transform of period 36 sequence [ -2, 3, 5, 1, -2, -8, -2, 1, 2, 3, -2, -3, -2, 3, 5, 1, -2, -2, -2, 1, 5, 3, -2, -3, -2, 3, 2, 1, -2, -8, -2, 1, 5, 3, -2, 0, ...].
a(n) = A164612(n) unless n=0. a(6*n + 1) = 4 * A233034(n). a(6*n + 2) = - A092848(n). a(6*n + 4) = 4 * A216046(n). a(6*n + 5) = A132179(n). - Michael Somos, Sep 05 2015
a(12*n - 1) = A230256(n). a(12*n + 2) = - A233034(n). a(12*n + 5) = A233037(n). a(12*n + 8) = A216046(n). - Michael Somos, Sep 05 2015
Convolution inverse of A164269. - Michael Somos, Sep 05 2015
EXAMPLE
G.f. = 1/q - 2 + 4*q - q^2 + 4*q^4 + q^5 + q^8 - 8*q^10 - q^11 - 8*q^13 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ -q^3] EllipticTheta[ 3, 0, q^3]^3 / (QPochhammer[ -q^9]^3 EllipticTheta[ 3, 0, q]), {q, 0, n}]; (* Michael Somos, Sep 05 2015 *)
a[ n_] := SeriesCoefficient[ With[ {A = 1/q QPochhammer[ -q^9, q^18]^3 QPochhammer[ q^3, -q^3]}, A - 2 + 4 / A], {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)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^18 * eta(x^9 + A)^3 * eta(x^36 + A)^3 / (eta(x^2 + A)^5 * eta(x^3 + A)^7 * eta(x^12 + A)^7 * eta(x^18 + A)^9), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Aug 11 2009
STATUS
approved