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A139093 Expansion of phi(q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function. 4

%I

%S 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,

%T 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,

%U 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

%N Expansion of phi(q) * phi(-q^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).

%C a(n) is nonzero if and only if n is in A002479.

%H G. C. Greubel, <a href="/A139093/b139093.txt">Table of n, a(n) for n = 0..10000</a>

%H M. 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)^7 / (eta(q)^2 * eta(q^4)^3) in powers of q.

%F Euler transform of period 4 sequence [ 2, -5, 2, -2, ...].

%F 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.

%F G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k-1))^2 / (1 + x^(2*k)).

%F a(8*n + 5) = a(8*n + 7) = 0.

%F 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).

%e 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 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2], {q, 0, n}]; (* _Michael Somos_, Aug 29 2014 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^7 / (QPochhammer[ q]^2 QPochhammer[ q^4]^3), {q, 0, n}]; (* _Michael Somos_, Feb 18 2015 *)

%t 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 *)

%o (PARI) {a(n) = if( n<1, n==0, 2 * (-1)^(n\2) * sumdiv(n, d, kronecker( -2, d)))};

%o (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))};

%o (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 */

%Y Cf. A002479, A033715, A033761, A082564, A112603, A125095, A133692.

%K sign

%O 0,2

%A _Michael Somos_, Apr 08 2008

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Last modified May 31 22:45 EDT 2020. Contains 334756 sequences. (Running on oeis4.)