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A139093
Expansion of phi(q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function.
4
1, 2, -2, -4, 2, 0, -4, 0, 2, 6, 0, -4, 4, 0, 0, 0, 2, 4, -6, -4, 0, 0, -4, 0, 4, 2, 0, -8, 0, 0, 0, 0, 2, 8, -4, 0, 6, 0, -4, 0, 0, 4, 0, -4, 4, 0, 0, 0, 4, 2, -2, -8, 0, 0, -8, 0, 0, 8, 0, -4, 0, 0, 0, 0, 2, 0, -8, -4, 4, 0, 0, 0, 6, 4, 0, -4, 4, 0, 0, 0, 0, 10, -4, -4, 0, 0, -4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, -2, -12, 2, 0, -8, 0
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is in A002479.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3) in powers of q.
Euler transform of period 4 sequence [ 2, -5, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112603.
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k-1))^2 / (1 + x^(2*k)).
a(8*n + 5) = a(8*n + 7) = 0.
a(n) = (-1)^n * A082564(n). a(2*n) = A133692(n). a(2*n + 1) = 2 * A125095(n). a(4*n) = a(8*n) = A033715(n). a(8*n + 1) = 2 * A112603(n). a(8*n + 3) = -4 * A033761(n).
EXAMPLE
G.f. = 1 + 2*q - 2*q^2 - 4*q^3 + 2*q^4 - 4*q^6 + 2*q^8 + 6*q^9 - 4*q^11 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^7 / (QPochhammer[ q]^2 QPochhammer[ q^4]^3), {q, 0, n}]; (* Michael Somos, Feb 18 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 (-1)^Quotient[n, 2] Sum[ JacobiSymbol[ -2, d], {d, Divisors @ n}]]; (* Michael Somos, Feb 18 2015 *)
PROG
(PARI) {a(n) = if( n<1, n==0, 2 * (-1)^(n\2) * sumdiv(n, d, kronecker( -2, d)))};
(PARI) {a(n) = my(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3), n))};
(Magma) A := Basis( ModularForms( Gamma1(32), 1), 105); A[1] + 2*A[2] - 2*A[3] - 4*A[4] + 2*A[5] - 4*A[7] + 2*A[9] + 6*A[10] - 4*A[12] + 4*A[13] + 4*A[16]; /* Michael Somos, Aug 29 2014 */
KEYWORD
sign
AUTHOR
Michael Somos, Apr 08 2008
STATUS
approved