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 A124863 Expansion of 1 / chi(q)^12 in powers of q where chi() is a Ramanujan theta function. 2

%I

%S 1,-12,78,-376,1509,-5316,16966,-50088,138738,-364284,913824,-2203368,

%T 5130999,-11585208,25444278,-54504160,114133296,-234091152,471062830,

%U -931388232,1811754522,-3471186596,6556994502,-12222818640,22502406793

%N Expansion of 1 / chi(q)^12 in powers of q where chi() is a Ramanujan theta function.

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

%H M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">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 q^(-1/2) * (k * k') / 4 in powers of q where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.

%F Expansion of q^(-1/2) * (eta(q) * eta(q^4) / eta(q^2)^2)^12 in powers of q.

%F Euler transform of period 4 sequence [ -12, 12, -12, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = f(t) where q = exp(2 pi i t). - Michael Somos, Jul 22 2011

%F G.f.: Product_{k>0} (1 + (-x)^k)^12 = Product_{k>0} (1 + x^(2*k - 1))^12.

%F a(n) = (-1)^n * A022577(n). Convolution inverse of A112142. Convolution square is A100130.

%e 1 - 12*x + 78*x^2 - 376*x^3 + 1509*x^4 - 5316*x^5 + 16966*x^6 - 50088*x^7 + ...

%e q - 12*q^3 + 78*q^5 - 376*q^7 + 1509*q^9 - 5316*q^11 + 16966*q^13 - 50088*q^15 + ...

%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ ((1 - m) m/16/q)^(-1/2), {q, 0, n}]] (* Michael Somos, Jul 22 2011 *)

%t a[ n_] := SeriesCoefficient[ Product[1 + q^k, {k, 1, n, 2}]^12, {q, 0, n}] (* Michael Somos, Jul 22 2011 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A)^2)^12, n))}

%Y Cf. A022577, A100130, A112142.

%K sign

%O 0,2

%A Michael Somos, Nov 10 2006

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