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A218171 Expansion of f(x^11, x^13) - x * f(x^5, x^19) in powers of x where f(, ) is Ramanujan's general theta function. 4
1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 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, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 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, 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).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^7, b = x. - Michael Somos, Nov 09 2014
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Quintuple Product Identity
FORMULA
Expansion of f(x^3, x^5) * chi(-x) in powers of x where f(, ) is Ramanujan's general theta function and chi() is a Ramanujan theta function.
G.f.: Sum_{k in Z} x^(12*k^2 + k) - x^(12*k^2 + 7*k + 1).
a(n) = A010815(2*n) for all n in Z.
EXAMPLE
G.f. = 1 - x - x^6 + x^11 + x^13 - x^20 - x^35 + x^46 + x^50 - x^63 - x^88 + ...
G.f. = q - q^49 - q^289 + q^529 + q^625 - q^961 - q^1681 + q^2209 + q^2401 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, If[ OddQ[ DivisorSigma[ 0, 48 n + 1]], JacobiSymbol[ 6, Sqrt[48 n + 1]], 0]]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] + QPochhammer[ q]) / 2, {q, 0, 2 n}]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] (QPochhammer[ q^2]^3 / QPochhammer[ q]^2/ QPochhammer[ q^4] + 1) / 2, {q, 0, 2 n}]; (* Michael Somos, Nov 09 2014 *)
PROG
(PARI) {a(n) = my(m); if( issquare(48*n + 1, &m), kronecker(6, m), 0)};
(PARI) {a(n) = my(m); if( n<0, 0, m = 2*n; polcoeff( eta(x + x * O(x^m)), m))};
CROSSREFS
Sequence in context: A113052 A369967 A256432 * A362130 A361018 A232714
KEYWORD
sign
AUTHOR
Michael Somos, Oct 22 2012
STATUS
approved

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Last modified June 23 08:04 EDT 2024. Contains 373629 sequences. (Running on oeis4.)