A260486 Expansion of phi(-x)^2 * phi(-x^6) / phi(-x^3) in powers of x where phi() is a Ramanujan theta function.
1, -4, 4, 2, -4, 0, 2, 0, -4, 0, 0, 8, -2, 0, 0, 0, -4, -8, 0, 8, 0, 0, 8, 0, -2, -4, 0, -2, 0, 0, 0, 0, -4, -4, 8, 0, 0, 0, 8, 0, 0, -8, 0, 8, -8, 0, 0, 0, -2, -4, 4, 4, 0, 0, -2, 0, 0, -4, 0, 8, 0, 0, 0, 0, -4, 0, 4, 8, -8, 0, 0, 0, 0, -8, 0, 2, -8, 0, 0, 0
Offset: 0
Keywords
Examples
G.f. = 1 - 4*x + 4*x^2 + 2*x^3 - 4*x^4 + 2*x^6 - 4*x^8 + 8*x^11 - 2*x^12 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^2 EllipticTheta[ 4, 0, x^6] / EllipticTheta[ 4, 0, x^3], {x, 0, n}]; a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ x]^2 QPochhammer[ -x^3] / QPochhammer[ x^3], {x, 0, n}]; a[ n_] := If[ n < 1, Boole[n == 0], -4 I^(n-1) Sum[ {1, I, -1/2, I, 1, -I/2}[[Mod[d, 6, 1]]] KroneckerSymbol[ -2, n/d], {d, Divisors[ n]}]]; a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, -x]^2* EllipticTheta[3, 0, -x^6]/EllipticTheta[3, 0, -x^3], {x, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Mar 17 2018 *) -
PARI
{a(n) = if( n<1, n==0, -4 * I^(n-1) * sumdiv(n, d, [-I/2, 1, I, -1/2, I, 1][d%6+1] * kronecker(-2, n/d)))}; -
PARI
{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -4 * I^(n-1) * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, I, p==3, 1-e/2, p%8 > 4, !(e%2), e+1)))}; -
PARI
{a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^6 + A)^3 / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^12 + A)), n))};
Comments