A128581 - OEIS

%I #20 Mar 12 2021 22:24:44

%S 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,

%T -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,

%U -2,-2,0,2,2,-1,0,2,0,0,0,-4,0,1,2,0,-3,0,-4,0

%N 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.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A128581/b128581.txt">Table of n, a(n) for n = 1..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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.

%F 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, ...].

%F 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).

%F a(n) = -(-1)^n * A115660(n). a(2*n) = A115660(n). a(2*n + 1) = A128580(n).

%F abs(a(n)) = A000377(n) = A192013(n). - _M. F. Hasler_, May 07 2018

%e G.f. = q + q^2 - q^3 - q^4 - 2*q^5 - q^6 + 2*q^7 + q^8 + q^9 - 2*q^10 - ...

%t a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ #, 8] KroneckerSymbol[ n/#, 3] &]]; (* _Michael Somos_, Nov 15 2015 *)

%t 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 *)

%t 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 *)

%o (PARI) {a(n) = -(-1)^n * sumdiv(n, d, kronecker(d, 8) * kronecker(n/d, 3))}

%o (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)}

%o (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

%Y Cf. A115660, A128580.

%K sign,mult

%O 1,5

%A _Michael Somos_, Mar 11 2007