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%I #22 Sep 08 2022 08:46:01
%S 5,4,4,-320,4,2504,-320,-9600,4,25924,2504,-58560,-320,114248,-9600,
%T -200320,4,334088,25924,-521280,2504,768000,-58560,-1119360,-320,
%U 1565004,114248,-2099840,-9600,2829128,-200320,-3694080,4,4684800
%N Expansion of phi(x)^2 * (5 * phi(-x)^8 + 64 * x * psi(-x)^8) in powers of x where phi(), psi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A204372/b204372.txt">Table of n, a(n) for n = 0..10000</a>
%H Michael 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)^4 * eta(q^2)^2 * (5 * eta(q)^8 / eta(q^4)^4 + 64 * q * eta(q^4)^4 ) in powers of q.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4*t)) = 2048 (t/i)^5 g(t) where q = exp(2*Pi*i*t) and g(t) is the g.f. for A050468.
%F G.f.: 5 + 4 * Sum_{k>0} (-1)^(k-1) * (2*k - 1)^4 * x^(2*k - 1) / (1 - x^(2*k - 1)).
%F a(n) = 4 * A050456(n) if n>0.
%e G.f. = 5 + 4*x + 4*x^2 - 320*x^3 + 4*x^4 + 2504*x^5 - 320*x^6 - 9600*x^7 + 4*x^8 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 (5 EllipticTheta[ 4, 0, q]^8 + 4 EllipticTheta[ 2, Pi/4, q^(1/2)]^8), {q, 0, n}]; (* _Michael Somos_, May 03 2015 *)
%t a[ n_] := If[ n < 1, 5 Boole[n == 0], 4 DivisorSum[ n, #^4 KroneckerSymbol[ -4, #] &]]; (* _Michael Somos_, May 04 2015 *)
%o (PARI) {a(n) = if( n<1, 5 * (n==0), 4 * sumdiv( n, d, d^4 * kronecker( -4, d)))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A)^2 * (5 * eta(x + A)^8 / eta(x^4 + A)^4 + 64 * x * eta(x^4 + A)^4 ), n))};
%o (Magma) A := Basis( ModularForms( Gamma1(4), 5), 34); 5*A[1] + 4*A[2] + 4*A[3]; /* _Michael Somos_, May 04 2015 */
%Y Cf. A050456, A050468.
%K sign
%O 0,1
%A _Michael Somos_, Jan 14 2012
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