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A128581
Expansion of (phi(q^2) * phi(-q^3) - phi(-q) * phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.
9
1, 1, -1, -1, -2, -1, 2, 1, 1, -2, -2, 1, 0, 2, 2, -1, 0, 1, 0, 2, -2, -2, 0, -1, 3, 0, -1, -2, -2, 2, 2, 1, 2, 0, -4, -1, 0, 0, 0, -2, 0, -2, 0, 2, -2, 0, 0, 1, 3, 3, 0, 0, -2, -1, 4, 2, 0, -2, -2, -2, 0, 2, 2, -1, 0, 2, 0, 0, 0, -4, 0, 1, 2, 0, -3, 0, -4, 0
OFFSET
1,5
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 eta(q^2)^3 * eta(q^3) * eta(q^12) * eta(q^24) / (eta(q) * eta(q^6)^2 * eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ 1, -2, 0, -2, 1, -1, 1, -1, 0, -2, 1, -2, 1, -2, 0, -1, 1, -1, 1, -2, 0, -2, 1, -2, ...].
Multiplicative with a(2^e) = -(-1)^e if e>0, a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1)(-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
a(n) = -(-1)^n * A115660(n). a(2*n) = A115660(n). a(2*n + 1) = A128580(n).
abs(a(n)) = A000377(n) = A192013(n). - M. F. Hasler, May 07 2018
EXAMPLE
G.f. = q + q^2 - q^3 - q^4 - 2*q^5 - q^6 + 2*q^7 + q^8 + q^9 - 2*q^10 - ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ #, 8] KroneckerSymbol[ n/#, 3] &]]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q^3] - EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^6])/2, {q, 0, n}]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q] QPochhammer[ q^24] QPochhammer[ q^4, q^8] QPochhammer[ q^3, -q^3], {q, 0, n}]; (* Michael Somos, Nov 15 2015 *)
PROG
(PARI) {a(n) = -(-1)^n * sumdiv(n, d, kronecker(d, 8) * kronecker(n/d, 3))}
(PARI) {a(n) = my(A = x * O(x^n)); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^6 + A)^2 * eta(x^8 + A)), n)}
(PARI) A128581(n)={prod(i=1, matsize(n=factor(n))[1], if(12>n[i, 1]%24, if(bittest(12, n[i, 1]), (-1)^(n[i, 1]+n[i, 2]-1), if(bittest(n[i, 2], 0)&&n[i, 1]%6>1, -n[i, 2]-1, n[i, 2]+1)), !bittest(n[i, 2], 0)))} \\ a(p^e) as given in formula. - M. F. Hasler, May 07 2018
CROSSREFS
Sequence in context: A366461 A261122 A115660 * A190611 A000377 A192013
KEYWORD
sign,mult
AUTHOR
Michael Somos, Mar 11 2007
STATUS
approved