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A232714 Expansion of f(-x, -x^6) in powers of x where f is Ramanujan's two-variable theta function. 6
1, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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
Euler transform of period 7 sequence [ -1, 0, 0, 0, 0, -1, -1, ...].
G.f.: Sum_{k in Z} (-1)^k * x^(k * (7*k + 5) / 2).
G.f.: Product_{k>0} (1 - x^(7*k-6)) * (1 - x^(7*k-1)) * (1 - x^(7*k)).
a(3*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.
Convolution inverse of A195849.
abs(a(n)) = A274179(n). - Michael Somos, Jan 28 2017
a(n) = -(1/n)*Sum_{k=1..n} A284363(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
EXAMPLE
G.f. = 1 - x - x^6 + x^9 + x^19 - x^24 - x^39 + x^46 + x^66 - x^75 - x^100 + ...
G.f. = q^25 - q^81 - q^361 + q^529 + q^1089 - q^1369 - q^2209 + q^2601 + q^3721 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ SeriesCoefficient[ QPochhammer[ q, q^7] QPochhammer[ q^6, q^7] QPochhammer[ q^7], {q, 0, n}];
PROG
(PARI) {a(n) = my(m); if( issquare( 56*n + 25, &m), (-1)^round( m / 14), 0)};
CROSSREFS
Sequence in context: A218171 A362130 A361018 * A274179 A194675 A117964
KEYWORD
sign
AUTHOR
Michael Somos, Nov 28 2013
STATUS
approved

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Last modified June 20 12:17 EDT 2024. Contains 373527 sequences. (Running on oeis4.)