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A228745
Expansion of (phi(q)^4 + 7 * phi(-q)^4) / 8 in powers of q where phi() is a Ramanujan theta function.
2
1, -6, 24, -24, 24, -36, 96, -48, 24, -78, 144, -72, 96, -84, 192, -144, 24, -108, 312, -120, 144, -192, 288, -144, 96, -186, 336, -240, 192, -180, 576, -192, 24, -288, 432, -288, 312, -228, 480, -336, 144, -252, 768, -264, 288, -468, 576, -288, 96, -342, 744
OFFSET
0,2
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
a(n) = -6 * b(n) where b() is multiplicative with a(0) = 1, b(2^e) = -4 if e>1, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 1/2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A228746.
G.f.: ( (Sum_{k in Z} x^k^2)^4 + 7 * (Sum_{k in Z} (-x)^k^2)^4 ) / 8.
a(2*n) = A004011(n). a(2*n + 1) = -6 * A008438(n).
Convolution with A005875 is A003781.
EXAMPLE
G.f. = 1 - 6*q + 24*q^2 - 24*q^3 + 24*q^4 - 36*q^5 + 96*q^6 - 48*q^7 + 24*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + 7 EllipticTheta[ 4, 0, q]^4) / 8, {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( (A^4 + 7 * subst(A, x, -x)^4) / 8, n))};
(Magma) A := Basis( ModularForms( Gamma0(4), 2), 51); A[1] - 6*A[2]; /* Michael Somos, Aug 21 2014 */
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 02 2013
STATUS
approved