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A226559 Expansion of f(-x^1, -x^7) * f(-x^2, -x^6) / (f(-x^3, -x^5) * f(-x^4, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function. 3
1, -1, -1, 2, 1, -2, -2, 2, 4, -4, -5, 6, 6, -7, -9, 10, 13, -15, -17, 20, 21, -25, -28, 32, 39, -43, -49, 56, 60, -69, -78, 86, 101, -112, -125, 142, 153, -172, -192, 212, 241, -266, -295, 328, 357, -397, -438, 482, 540, -592, -652, 720, 781, -862, -946 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
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 (f(-x^1, -x^7) / f(-x^3, -x^5)) * (psi(-x^2) / phi(-x^4)) in powers of x where psi(), phi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [-1, -1, 1, 2, 1, -1, -1, 0, ...].
Given g.f. A(x) then B(q) = q^3 * A(q^4) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v - u^2)^3 - 4 * u^2 * v^3 * (2*v - u^2) * (2 + v^2 - u^2*v).
a(n) = -A092869(2*n + 1) = A230534(2*n + 1).
EXAMPLE
G.f. = 1 - x - x^2 + 2*x^3 + x^4 - 2*x^5 - 2*x^6 + 2*x^7 + 4*x^8 - 4*x^9 + ...
G.f. = q^3 - q^7 - q^11 + 2*q^15 + q^19 - 2*q^23 - 2*q^27 + 2*q^31 + 4*q^35 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{0, 1, 1, -1, -2, -1, 1, 1}[[ Mod[k, 8] + 1]], {k, n}], {x, 0, n}];
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n))^[ 0, 1, 1, -1, -2, -1, 1, 1][k%8 + 1]), n))};
CROSSREFS
Sequence in context: A208217 A009213 A072209 * A060169 A117958 A113401
KEYWORD
sign
AUTHOR
Michael Somos, Jun 10 2013
STATUS
approved

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Last modified March 1 11:36 EST 2024. Contains 370432 sequences. (Running on oeis4.)