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A164268
Expansion of f(q^3) * phi(q^3)^3 / (q * f(q^9)^3 * phi(q)) in powers of q where f(), phi() are Ramanujan theta functions.
5
1, -2, 4, -1, 0, 4, 1, 0, 0, 1, 0, -8, -1, 0, -8, 0, 0, 4, 1, 0, 16, -2, 0, 16, 0, 0, -4, 2, 0, -32, -3, 0, -32, 1, 0, 8, 4, 0, 56, -4, 0, 56, 1, 0, -16, 4, 0, -96, -6, 0, -92, 1, 0, 24, 5, 0, 160, -8, 0, 152, 1, 0, -40, 8, 0, -252, -10, 0, -240, 2, 0, 64, 11
OFFSET
-1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
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(3*n) = 0 unless n=0. a(3*n - 1) = A062244(n). a(3*n + 1) = 4 * A128111(n).
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))};
KEYWORD
sign
AUTHOR
Michael Somos, Aug 11 2009
STATUS
approved